basic information again and again. So last time we covered what we covered trigonometric t ratios, basic t ratios and heights and distances. Today we are going to take up identities. So that's what we are now going to do. So again, if you see this is the deleted portion. So you know in trigonometry, how much is a trigonometric score or weightage in the board, you know already. So let me just select a pointer first. Yeah. So these are eliminated or these are not there anymore, heights and distance, no elimination trigonometric identities, trigonometric ratios of complementary angles are not there. Okay, so hence we will not be dealing with them. Okay, now this is again to revisit the pattern of the paper. So that, you know, now we are doing it multiple number of times. That doesn't mean that it is going to be, you know, you never know, board can come up with a new pattern also, but mostly unlikely, but you know, till anything comes out or any changes are made, we will assume that this is what is going to be for the board exams. Now, so what are the, what is the pattern again? So you know that there are 80 marks paper, this is 80 marks paper for the regular people. For the autonomous people, we have, you know, I have the declared from your school side, I have to just confirm with the school as well. But I think last time they had a 50 marker paper, but it will not be a 50 marker, it will be either 70 or 80, mostly 80. So we'll have to just confirm. But anyways, the pattern, the type of questions are going to be similar. Now, so there are 16, one marker, one mark questions. Again, if I have to, you know, start, right, one more question, how many? So you have to be very, very thorough 16. And there are how many four case studies of four marks each. So we have four, you can, you can assume that also. So let's say 16, 16, one marker again, one marks in four, four case studies, there will be five questions, you'll have to do four out of five, right? So this is the pattern. Now, this is 16. So hence this is also 16, 16. So total is 32. So part A is 32 marks. Part B will obviously be off how much? 48 marks, right? So 48 marks breakup is what? So there are very short answer type questions of two marks. So if you see two marks, six questions, six questions, so six questions into two marks is equal to 12 marks, three questions or seven questions of three marks. So is equal to 21 marks and three question of five marks is 15. So this is nothing but 48. So this is the breakup, you will get some internal options in few of the questions, right? But you can't skip any particular topic. In one particular topic itself, there will be two questions given. Okay, so this was the structure. Now let's come to identities. So today we are going to deal with identities and we'll be solving all the previous year questions, at least the previous year, the sample number of this year and 2018-19. So you get fair enough idea of what type of questions are being asked. So hence in grade 10, we have only three types of identities which are called Pythagorean identities as well. So sine square theta plus cos square theta is one. So this is one, one plus tan square theta is equal to second square theta and one plus cot square is cosecant. So CSE is nothing but cosecant square theta. Okay, so proves you all know, I think. Yeah, so if you have to prove it, what do you need to do? You just need to, you know, take a triangle and yeah. So basically, okay, let me let me do it here as well. So now that I have some free space here, I've deliberately kept it. So we are going to first prove and then we'll be talking of some different, some various manifestation of the same identities. Okay, so this is P perpendicular B and hypotenuse H. This is theta, let this triangle be ABC. Okay, and clearly sine square theta, what is sine theta here? So if you see sine theta is equal to E by H and cos theta is equal to B by H. So square both sides are basically sine square theta plus cos square theta will be equal to P square upon H square plus B square upon H square is equal to P square plus B square upon H square. Right, so hence this will be nothing but P square plus B square by Pythagoras theorem is H square upon H square. Hence one. So this is the most basic identity you'd have learned and though they will not ask for the proof, but yeah, you should also have this in mind. Now, what important about this identity will be different manifestations, what do I mean? So sine square theta plus, and this is the basis of the questions which will be there. So sine square plus cos square theta is one, which can be very easily written as sine square theta is equal to one minus cos square theta. Now all these forms and manifestations must be clear in your head. This is where lots of questions on board have been formulated. So we'll solve a few of them anyways and I'll show you how this is so relevant. So hence, this can be very well written as, so this can be written as sine theta into sine theta is equal to one minus cos theta into one plus cos theta and this is very, very important manifestation. So what is this?  This is very, very important way of writing it. So this use of this is so many times. So you remember this. Also the same thing can be written as cos square theta is equal to one minus sine square theta.  So these are different, different types of manifestations. So you know the questions will be there where you have to replace them like that. So please remember. So this is how you can treat one, one identity is one identity is same as so many and the other important point is wherever you have, you are seeing this one. So one, one can be written as sine square theta plus cos square theta. There will be many places where you have to apply this. Yeah, I request all the students to join on time because what happens is the moderator gets distracted. So hence all students should be joining on time. If you are struggling, you can immediately reach out to any of the representatives groups, but coming on time will be very, very crucial. Okay, so please do that. Do not wait for anything. Just join in before 3.30. So we have now put the timings also such that the link link is active, active around 3.20. So hence join and all this. Okay, so coming back to so this is clear. So the you have to keep this in mind.  So one, this brings these three together. So keep that in mind. Now, next one is again, very important. Again, it is called a Pythagorean identity, trigonometric identity one plus tan square theta, sequence square theta. So there will not be much of a problem to prove it. So you can do again. So let us say this is A to B, C, and this is theta. This is 90 degrees. This is base. This is hypotenuse. This is perpendicular. So one plus tan square theta, one plus tan square theta is equal to one plus tan theta is opposite by adjacent. And then there is a square. So this is one plus p square upon B square, which is B square plus p square upon B square.  So this is the proof. Okay, so this is the proof. Now again, manifestations are important, not only the identity, but how to write the identities multiple number of or in multiple ways. So one plus tan square theta can be written as secant square theta. And here is the game. The game is like this. So you can write it. You can write secant square theta minus tan square theta is equal to one. And now you must be aware that this can be written as secant theta minus tan theta into secant theta plus tan theta equals one. That means secant theta minus tan theta is reciprocal of reciprocal of secant theta plus tan theta and vice versa. Understood. So secant theta minus tan theta is important, right? Secant theta minus tan theta can be expressed like this. Alternatively, secant theta minus, sorry, plus tan theta reciprocal is one upon secant theta minus tan theta. So just by changing the sign of tan, right, you can get the reciprocal. So many a times this will be applicable again. So keep this in mind. Another way of representing this would be another way. Oh, sorry. Yeah. Wait. Chat. Where did the chat thing go? Okay. So yeah, another way of what was I saying? I was saying another way of representing this would be this. What you can write this as secant square theta minus one is equal to tan square theta. Okay. So this is going to be important. Now secant square theta minus one is tan square theta. Again, you can write this as you can write this as secant theta minus one secant theta plus one is equal to tan square theta. And again, you can see that secant theta minus one upon tan theta can be written as tan theta upon secant theta plus one. Right. So and vice versa, meaning so secant theta plus one upon tan theta is equal to tan theta upon secant theta minus one. So you must remember these manifestations also. Okay, next.  So this is written in short form cosecant c o c s c. Okay. Now one plus cot square theta one plus cot square theta is nothing but one plus cot theta cot theta is adjacent by opposite. So B by P whole square, which is nothing but one plus B square by P square, which is nothing but B P square plus B square by P square, which is nothing but h square by P square, which is cosecant square theta. So this is the proof. And then again, same thing what we did for the last identity one plus cot square theta is equal to cosecant square theta. So again, you can write cosecant square theta minus cot square theta is equal to one. So all these different varieties must be in your mind quick. So you should be able to relate to solve the problem quick, right? So once this can be written as cosecant theta minus cot theta will be a reciprocal of cosecant theta plus cot theta. So keep this in mind again, you don't need to mug up, but you meet you please keep in mind that okay, this is how these identities or these ratios can be expressed. Okay, similarly, you can do the reverse. So cosecant theta plus cot theta is reciprocal of cosecant theta minus cot theta, right? And the final one, what we did the other way on the secant walla identity as well cosecant square theta minus, or this we did minus one, minus one is equal to cot square theta. Okay, so which is nothing but cosecant theta minus one by cot theta I'm writing directly is equal to cot theta. You can relate by the previous example cosecant theta plus one correct. And the other way round, which is cot theta or cosecant theta or whichever way you can write like that also cosecant theta minus one is equal to cosecant theta plus one by cot theta. Okay, is that okay? Is that okay? So this is the basic or basic understanding of identity. So everything you will be doing now will be a combination of whatever we just did nothing more, nothing less. Okay, so here is a grid which you don't need to mug up, but you must be having it with you the slides will be with you. So hence you can go through what we have done is all the trigonometric identities have been expressed in terms of the other. So you can you can ignore the minus part here because you are not dealing with any angle greater than 90. So in all the plus minus you see you just can keep only plus minus you can ignore for the time being. Okay, so this is how sin theta is related to sin theta, how sin theta is related to cos theta, how sin theta is related to tan theta, how sin is related to cosecant, secant and cot like that. So all interrelationships are there. So you can keep as a you know, some kind of a handy result, you don't need to mug anything from here. And yeah, so let's start right away with this. So as you know, there will be one marker. If there is any identity question, there will be either one can be two can be three as well. So one, two, three, there will not be I think any question for five marker because five marker usually would be for height and distance. So you can expect any question one, two, three, one marker, two marker, three marker. So I think minimum of three marks for off identities would definitely be there. Maximum of four marks not beyond that, I believe. But anyways, so or maximum here, anyways will be five. So there will be, as you know, there are how many questions from tignometry total is 12 marks. So there is one plus one, two, one marker, one, two marker, one, three marker and one, five marker. So total is four plus three, seven, seven plus five, 12. So this is the structure or tignometry. So in out of these two, one here, one will be off. Dux, you're saying something, Dux. No, sir. It was a mistake. It's just opening the chat. No problem. So one plus one. And so one could be tignometry. Two could be tignometry. Three could be tignometry. Okay. So you know the, sorry, tignometry identity. So at max five marks for tignometry identities. Okay. So let's begin guys. So here is the first question. This is given in the sample paper, which is published by CBSE. So it should not be a big deal. So could you find X plus Y or all of you have done already? So three, very good. So how would I write it? And you must also need to check the marking scheme, guys. So in marking scheme, the previous class, I forgot to tell you that in the heights and distance question, there is one mark for diagram only. Okay. One mark is on diagram. So don't miss that diagram, right? One mark on diagram. So draw it very, very neatly and, and then every other step is half a mark, you know, worth half a month. So every step you write, you'll get half a mark with that. Though people do not go into that deeper detail, but then once, you know, while correcting it, but definitely since it is given that diagram will have one mark. So you will get one mark if you have drawn the diagram correctly. Okay. So keep that in mind. So last time we could not discuss that. Now in case of trigonometric identities, many a times the question comes whether I have, I can do both LHS, RHS and all that. So usually the practice is to start from LHS or go to RHS or to start from RHS and go to LHS whichever way, but it's not, you know, very sacrosanct, so to say, but let's say if you are stuck totally, you can always do that where you can, you know, you start from both sides and prove eventually. But the questions which are there, you'll see the previous year questions also in none of these questions you will be, you know, finding that difficult to prove. Okay. So very clearly, so X plus Y is given nothing to do.  Now here there's an important thing. Now many a times you would maybe you have forgotten to write the reason. Let's say this reason because sine square theta plus cos square theta is equal to one. Let's say you have missed it. Okay. Now many a times people do what they directly jump from here and they write three answer as three. Instead of that, I would say even if you know the answer, try to write like in this form two times one plus one, which clearly indicates that you know that this one has to come here. Is that understood? Give me a thumbs up if you understood the point.  But if you have forgotten or let's say this is a good practice to, you know, write this step two times one plus one like that. Okay. Fair enough. So this is answer is three. All of you know this next. So the value of this is and you know, your entire trigonometry one is the number to chase, right? One. One is the number to chase. And this is very, very simple again. So you'll write sine square theta plus one upon secant square theta, right? Why? Because one plus tan square theta is equal to secant square theta. We just learned that. And hence it is sine square theta plus cos square theta. Why? Because one by secant theta is equal to cos theta. And hence it is one. Okay. Done. Next. This one, again, number to chase, you know, the favorite number to chase here is one. These are all one marker. This was asked in 2020 board paper. Okay. So again, the value of this is nothing but one plus tan square theta times times one minus sine square theta, one minus sine square theta. If you have time, you can write a plus b a minus b, but this is a one marker. So you can avoid that. And then one plus tan square theta can be written as secant square theta. And this is, wait, somebody just joined. Okay. Anyway, so secant square theta plus, sorry, not plus into cos square theta is equal to one. Why? Within the brackets, you write secant theta into cos theta is equal to one. Done. Next. If sine a plus sine square a is one, then find the value of expression cos square a plus cos to the power 4 a. Let me know if you could do it. So what's the answer? Typing. So it is one without doubt. Okay. So how to do it? How to do it? So sine a plus sine square a. Now this is a mouth watering. Sine square a is the term. So sine a clearly is one minus sine square a, which is cos square a. So sine a itself is cos square a. Okay.  Isn't it? Why? Because, because we just proved that cos square a is equal to sine a. And this is already given one here. So done. Yes, Navya, please rejoin, but you can't hear. How do I, you can, anyone who cannot, who's facing any technical glitch can rejoin. Okay. Next, Next, one marker What is the value of, so you can see there are lots of ones around one, one, one, one, one very very easy so one marker will be very, very easy. So you have secured one out of 12 for short, so one by one plus cot square, so it is nothing but one upon one plus cot square is cos square theta plus one upon upon secant square theta is equal to sine square theta plus cos square theta is equal to one and you can always write your reasons that secant or sorry secant square theta is equal to one plus tan square theta and cosecant square theta is equal to one plus cot square theta like that okay so this is on one mark over now yes simplest form of one plus tan square a by one plus cot square a is writing reason compulsory you in one marker you can avoid or you can you know leave but for three markers and all yes please write wherever it is wherever you feel that you know wherever you are using anything okay now the thing is as I told you you have enough time lots of time so let's say in the first go you did not write or you missed it then in the second round when you are revising now so it's easy on the right hand side it will be enough space will be there just keep on writing the reasons in the next round when you are revising is it understood hence we divided the time accordingly so you have 15-20 minutes in the next round in fact more than that so always come back and keep writing wherever you think that there is some scope of writing the reason okay so it's yeah answer is tan square a and I don't need I don't think I need to prove it so one plus tan square is directly secant square a and we have us divided by one plus cot square is cosecant square a which is tan square a or you can also write one more step that is sine square a by cos square a okay so which is nothing but sine by cos is tan square a done so one markers are pretty simple right so none of you should be losing any single mark here so what is our objective what is our objective guys just to remind you our objective what are we chasing what are we trying to get yes centum so everyone should be only chasing centum in centum get centum okay so centum yes so centum has to be there hundred marks is very doable okay next here now we are going to three marker identities which have been there in the previous year papers so here is the first one very very simple one I believe I think you will be able to do it and there could be multiple ways of solving the same one a chop one more piece of advice before you attempt now there is someone who is entering without a proper name okay so who's this guy just a minute people do not understand a simple fact sorry you can't let them enter okay yes guys what I was saying is before you attempt identity problem it is always advisable take birth check color whether the identity is true or not correct or not because many times what happens you will see that it is you know not true maybe maybe so hence it is always advisable so best for tan and quarter if it is there you can use 45 degrees and you can just a minute here who's this bugger unknown person random name random names people will not be entertained in the class okay I must know you okay prove that one so hence done Sujani has done anyone else on prove of shit also done very good and if you look at it very carefully guys it is nothing but we just discussed about see this has come in the board paper is it it you've just discussed about manifestation of this one I just this is based on this yes or no this is based on this part so cot square theta is equal to cosecant square theta minus one that's it and done so it is based on that manifestation only so what is this so I told you that cot square alpha is equal to cosecant square alpha minus one is it so hence I can write cot square alpha is equal to cosecant alpha minus one and cosecant alpha plus one okay so hence this is nothing but cot square alpha divided by cosecant alpha plus one is equal to cosecant alpha minus one so this implies course you can write cot or you can write here cot square alpha divided by this is cosecant alpha plus one or one plus cosecant alpha plus one is equal to cosecant alpha and prove right so this is again so if you in such questions do not try to reduce into cos and sign directly when you see it when you're seeing cosecant and caught clearly so this was this one has to be taken here like that and you could have got the clue oh correct is that understood any doubt how to approach identities yes I was saying you can also prove it take alpha is 45 degrees so cot 45 is one so one plus how much is it one plus two so cot square alpha so you can or you can take LCM and solve you can do that I told you there are multiple ways of doing it till you get the LHS equal to RHS you can do all sorts of permutation combination but again what I see in this we spent how many steps one two three four done so you know there you could adopt any method which is helping you no problems at all okay here next one this is very simple straightforward very very simple and straightforward just say done when done just say done when done okay quick all of you where are there are 90 people 90 plus people come on guys where is the energy boss centum clinic area energy you need energy energy energy high energy done done done done quick quick quick very good very good it's very much possible centum is within reach so hence everyone should get 100 on 100 no less if more I don't mind but not less okay yeah come on done everyone done yes even if you know even if you know the question even if you know the question solve it even if you know the question solve it solve properly solve and see whether you are writing and your you know thought processes are correct or not so I know most of the questions would be very easy you'd have done it whether in the pts or new mock exams which are writing wherever but there is no problem in doing once again okay so how to do it very very easy I would have taken so I'll start with LHS LHS is equal to tan square theta plus tan to the power 4 theta which is equal to tan square common theta 1 plus tan square theta isn't it now tan square theta itself can be written as secant square theta minus 1 and 1 plus tan square theta can be written as secant square theta and the job is done secant 4 theta minus secant square theta within this I will write since secant square theta is equal to 1 plus tan square theta done okay and you get three marks and you get closer to okay next so this is another variety where they are equating something to some third variable right P and Q are there so you have to simply substitute into LHS arrive at RHS is another way of testing identity you know your skills and identities okay so simple don't need to do anything substitute calculate and complete the process the best part about identities is that you know the end result so there is absolutely no way you can go wrong right so hence you know the end result improving questions so becomes much easier yep done so all done will say done so monish is done stress done arundati done showbita done Pranav done Aditi done Akshita done very good Anish also done Anike Srijani Vaishnav Surya Anirul Sharan great guys good Siddharth Karthik done very good Ananya Aryan very good nice Monish also done Monish Gouda done very good guys so simple again so what to do Q be very very careful what is being asked for secant theta plus cosecant theta is Q multiplied by P square so right sin theta plus cos theta whole squared this is P square minus 1 right so you have to arrive at 2p so 2p keep in mind is nothing but 2 times sin theta plus cos theta very good so what do I need to do also I have an expansion to do here so secant theta plus cosecant theta into sin square theta plus cos square theta plus 2 sin theta cos theta minus 1 this is the thing right simple so hence now first let's simplify this secant theta plus cosecant theta since it is three markers I'll write 1 plus 2 sin theta cos theta minus 1 and here I will write the reason sin square theta plus cos square theta equals 1 and then I will cancel 1 and this one so while you are canceling make sure that you know whatever it is being cancelled is also visible right so that the examiner knows what are you trying to do so this this is nothing but 2 times you can write now sin theta cos theta times 1 by cos theta plus 1 upon sin theta like that you can write or you can write directly as well no problems so it is 2 times sin theta cos theta by cos theta plus 2 times sin theta cos theta by sin theta so now so hence it is too common sin theta plus cos theta which is nothing but 2 times P fair enough any problem anyone wants all clear all good any any issues any glitches any any concern right so far so good so we are nearing closing closing in towards centrum okay so please maintain your you know pace and keep yourself calm as well okay here next one now this is again one of the manifestations which we talked about and you will be able to do it do it board paper question okay surya has done it very good monish agati siddharth very good so the hint is you have to divide the numerator and denominator with something okay very good so most of you have anyways done it so hint is you have to divide the numerator in such cases where you have secant and tan on the RHS and sin in cos on the left hand side so yes so simply divide numerator and denominator by cos so hence and you have to write also so you have to write dividing dividing numerator and denominator by cos theta okay you will get LHS is equal to sin by cos theta minus cos theta by cos theta now many a times it appears to be too too much work to write all that is obvious but my dear friends you have to make clear in the minds of the examiner that you know it so hence you have to walk that path so sin theta by cos theta minus 1 upon cos theta anyways it's a matter of just three months now from here on so it is nothing but you can see tan theta and then minus 1 plus secant theta and in the denominator we have 1 plus tan theta minus secant theta and now there is a catch yes now there is a catch again this one is going to rescue you out and since the thing is in the denominator so we'll only go by only go for that substitution in the denominator so what will you do you will write this as secant theta minus or plus tan theta minus 1 just rearranging the terms in the numerator divided by divided by so one in the denominator can be expressed as one in the denominator can be expressed as secant square theta minus tan square theta and this is the catch in this identity plus tan theta minus secant theta and you will see that the numerator and denominator will get a common term so what is this so secant theta plus tan theta minus 1 and now you can say secant theta minus tan theta into secant theta plus tan theta and extend it extend it and here take minus one common and write secant theta minus tan theta okay so what will you get you will get secant theta plus tan theta minus 1 divided by secant theta minus tan theta so this is if you would have seen you are you're already there in the you've got the destination wall apart okay and what is there in the common thing secant theta secant theta plus tan theta minus 1 and you can see numerator and denominator has a common factor when this entire thing and this entire thing will go and you get the LHS sorry RHS secant theta minus tan theta okay so the catch was yeah no what to do so there are you know one one indication was that RHS is secant tan so I have to divide by cos to get secant tan in the LHS as well and then one of the ones because there is three three terms and on the RHS there are two terms in the in the denominator that means you have to create a pair somewhere so I chose this to create a pair and then I could solve it okay clear clear guys right so this is three marker again so you can mark this as a star problem where some substitution was done and you know you just have a look before your exams again okay next so you know radical sign questions here you know what to do whenever there is a radical sign question you have to try to multiply and divide by conjugates right yep so this is very easy Surya is already done so whenever there is a square try to eliminate the square root sign and how can you eliminate the square root sign by completing the square method or you know not to complete in the square method making it perfect square somehow so most of you have already done easy problem folks easy I just tell me very honestly so far was there any question which was too hard any question which was very hard and oh my god trigonometry is like I am it's not my cup of tea I cannot do well in trigonometry oh I am scared of trigonometry right so you can see these are I have not fabricated these questions myself these are all board papers all previously asked questions or the sample papers which have been given by the board and mostly these are you know you have you must have done that in the school as well okay so how to do as I told you whenever you see a radical sign question radical sign meaning wherever this sign is there what to do you will try to and then on the right hand side there is no such sign so obviously I will do something to eliminate a square root sign and how to eliminate a square root sign no don't say rationalize because even if you multiply by the one plus sign a the result you know you will not get as a rational factor sin theta is not always rational okay so rationalization is a wrong term to use you basically do what multiply and divide by the conjugates what is conjugates 1 minus sin 8 of conjugate is 1 plus sin a like that right so hence what do we do you do this so LHS is equal to root over 1 plus sin a divided by 1 minus sin a and then multiply this with root over oh sorry or yes you can do that also instead of root over you could have done directly but I do not have a razor in front of me so a 1 plus sin a and divide by 1 plus sin a this is what I have done so which is nothing but root over 1 plus sin a whole square and in the denominator I will get 1 minus sin square a right which is nothing but root over let me write 1 plus sin a whole squared divided by cos square a so hence nothing but 1 plus sin a and the root is gone because I created perfect squares run and this is 1 upon cos a plus sin a by cos a sorry which is secant a plus tan a done and you can always write here sin square a plus cos square a is done guys why can't we just square on both sides see again Prisham the thing is usually we take this root again it's not wrong you can do that but usually the standard conventional methodologies to take one side and go to the other side okay so when you square it and try to prove will not be a wrong thing to do but again as I told you you know instead of taking risk let's take let's go by the convention we start from one side go to the other side Prisham clear so try avoid try avoiding you know doing something to the entire equation maybe okay next oh this is so very very simple this should be one marker I don't know why they have given us three marks soo sandhya Aryan soo sandhya you are late today as you had gone out what do you do out there is nothing to do outside cuts of corona virus there and there why do you go out oh okay I hope all well cool chalo solve it solve it solve it solve it solve it cot a minus cos a upon cot a plus cos a equals cosecant a minus one upon cosecant a plus one it's like ksribat alwa very easy question piece of cake isn't it trigonometry is easy in fact maths entirely is easy very easy subject everyone should study maths okay done everyone are you where is the energy you are only five 15 20 people are responding where are all watching india australia match huh oh that's over by i think india australia match is over no the test match are you watching the highlights vaishnav vaishnav got interested now okay over who won australia oh you are okay suya suya is playing game doesn't know he says okay chalo so let's come back to trigonometry cot a minus cos a by cot a plus cos a do so what what do you do okay who suggests what to do divide or something multiply divide yes or no yes so divide simply by cos and done so hence you write dividing a numerator and denominator of lhs by cos a you will get cot a by cos a minus one divided by cot a by cos a minus one sorry plus one and done so cot a by cos a anyways is cosecant a cosecant a minus one so one step cosecant a plus one and you have to mention but cot a is equal to um cos a by sin a is equal to cos a into cosecant a yes one step that's what i'm saying so they they have given in sample people no so anyways chalo no worries let's go to this one this is a sister of the first one so do i need to prove this no need same thing so what is the dividing thing here what will you divide it with what will you divide it with sin very good so divide simply by sin so you'll get tan a by sin a plus one by tan a by sin a minus one so they're just played with the signs and same thing so tan a by sin a is so you can always write sin a by cos a into sin a so i've written tan as this plus one and then so see brackets here and there to make sure that you know the examiner follows what you're trying to do is always beneficial so write like that minus one right so sin sin goes so this becomes sin sin goes so one upon cos a plus one by one upon cos a minus one so hence it is secant a plus one divide by secant a minus one hence proved not a big deal chalo you got so you can see this is a scanned copy of the board paper so you can see the texture of the background of the question also is reflecting just expand no don't expand yourself done so without just seeing the question done hey right even if you know it you should write akshita dash aditi right right right i can't oh no really done oh good does cbc plagiarize from rds or something i will tell those all these authors have been connected to one of the agency schools institutions like that okay done very simple very very simple so as daksha was mentioning expand yes expansion is great so let's expand so bahi disha may lhs is equal to expand so it is sin square theta plus cosecant square theta plus two times sin theta times go secant theta plus cos square theta plus secant square theta plus two times cos theta secant theta right now club so this is equal to sin square theta plus cos square theta and then cosecant square cocaine we will write this as plus cot square theta plus one and then two sin theta cos theta is simply two and then this is gone secant square colic in a tan square theta plus one and this is again two so hence this is one one so one plus one plus two plus two or one plus one also one plus one plus one plus tan square theta plus cot square theta so this entire thing is seven seven plus tan square theta and cot square theta and here write all the identities together because all of them are used so right sin square theta plus cos square theta is equal to one then write secant square theta is equal to one plus tan square theta and third cosecant square theta is equal to one plus cot square theta that's it done cool one more inch towards center next yeah so who's saying done dachsh and munish saying done what did you do akshita also says done that's why what did you do so we write cot as cos by sin and cosecant as one by sin and just convert everything into sin and cos and we simplify the two multiplication brackets and then you multiply it get a square minus b square at the top and now it will be sin into cos so did you did you uh uh sin and cos convert akshita what's your what's your process can you unmute and say akshita is there yes sir same thing cot a cosecant tan and secant if we convert everything to sin and cosec it is um i mean when you simplify it we get to oh is it okay or did you expand or not did you multiply did you multiply the two so you will get what three into three nine terms did you get that or no manu munish what is munish's process substituting cot as sin and cos and cosecant as one upon the same thing and then expand is that is that what you did yes sir any other any other method did anyone solve without expanding i was just curious to know you multiplied and then substituted okay is this can this sum be done without multiplication is that possible no yes or no anyway so one thing is very clear that we can definitely multiply and get that so hence instead of uh yeah expanded only so let's say if you expand what will you what will you get first is one plus tan a plus secant then you will get plus cot a then plus cot a tan a so you now know your fodder is here cot a tan a and one so that is two that is all others will get added up to zero okay never mind let's begin continue so cot a secant a and third expansion minus cosecant a minus cosecant a tan a and then minus cosecant a secant a perfect these are nine terms so count one two three four five six seven eight nine perfect perfect now so oh sorry now so you now know what what do you know this is one this is one so let's write one plus tan a plus secant a plus cot a i don't know the fate of all of this plus one plus cot a secant a what will this be cos a it is nothing but cosecant a yes or no am i right yep now this is minus cosecant a so you got the pair now cosecant tan a is minus secant a adjunct it and this is minus cosecant a secant a let it be like that so beta g beta beta g gone anything all any kuchar bachaya no so hence it is two plus tan a is sine a by cos a now i will change it then cot a is cos a by sine a minus what one upon cos a sine a and then done over games over why because this is two plus take sine and cos as common denominator and then here it is nothing but sine square a plus cos square a minus one which is zero this item is zero hence you can write two plus one minus one upon sine a cos a okay and all the reasons again bracket with tino identity all the three put together write it down yep so let me let me write it down sine square theta plus cos square theta is equal to one secant square theta minus oh that is not required i believe so this is not required and you can always write cosecant a into what was that it got killed cosecant into tan okay is equal to what secant a and the third one also you have to write like that just write fair enough done any problem so three into three nine terms multiplication and done okay this one oh this is same we are not going to eat this is again see there's a repetition question is repeated let's not do it this one do this even if you have done this seen this please for god's sake do it do properly as if you are viewing it for the first time akshita was done monish done two people come on josh josh josh energy pranav done nanniket gupta is done chart log come on daksh singh done very good come on guys as a hint how many of you want hint no one anyone wants hint oh aditya also medha arian aditya done aditya done others shristi done very good anushka shobhita very good very good moving towards centim good keep it up guys shrijeni done very good shrijeni what is the difficulty coefficient so far questions easy difficult crazy edc what is the survey so far e andrella says c crazy edc if is e difficult or crazy totally weird uh e all of you are e very good nice good going guys good going chalo chalo keep the momentum going and how to solve this so hint was simply i would touch one of the two terms not both see this is a beauty one minus cot theta can be converted into one minus tan and vice versa so what i'll do is uh i will do this part so tan theta um let's say yes or one minus cot theta can be written as one upon one upon cot theta minus tan theta by cot theta can i do that below or this is required uh yes some easy some are okay are in single yes any other root so what i'll do is i will touch only one instead of both or what i can do is tan one by tan is cot and yes guys even in terms of sign or you you have changed into sign and cos yeshika yeah so every cot into tan yeah that will be better actually yes basically yes that is better to do so what i'll do is i have done yeah tan cot cot tan may change to that's better yes correct so tan upon one minus cot and cot can be written as so let me rewrite this part this is one upon tan and this is one minus tan theta very good oh this cot also will be changed so let me erase everything let me erase erase no gajpach all clear very clear neat and so if you're let's say you got stuck somewhere and you have to delete anything don't paint there just simply you know strike it off and move forward so tan theta and this is one minus one upon tan theta that's what i was saying and this is one upon tan theta by one minus tan theta and this is will this will give you one minus tan theta denominator so if you see this is tan square theta this tan theta will go up here and this will become tan square tan theta minus one and this can be written as um yeah this can be written as plus one upon tan theta minus tan square theta did i do the correct thing uh tan theta let it be tan theta sorry one by tan don't multiply sure yes yes so tan theta one minus tan theta so that i can take common right now what do i do i take tan theta minus one as common one upon tan theta minus one as common so what do i get i get tan okay just check the calculation tan theta goes up and square is better tan theta minus one plus one upon tan theta one minus tan theta perfect so i take let's say one upon one minus tan theta is common so what am i left with i'm left with minus tan square theta and what what's here plus one upon tan theta am i right so far so good yes or no fine and cos seems friendly you can do that no problem so this is nothing but you will get one upon one minus tan theta and then if you rewrite it this is tan theta in the denominator and in the numerator you will get one minus tan cube theta i hope this makes sense correct so what so now one minus tan theta will go from the bottom you know denominator so it is nothing but one minus tan cube can be written as one minus tan theta one plus tan square theta plus tan theta a q minus b q form divided by one minus tan theta into tan theta so this will go now simply so this is one by tan theta is how much one by tan theta is cot theta correct so hence i am getting watch out anyways one plus tan square theta can be written as secant square theta will that help will that help so now here you can reduce sin and cos everything so that you'll get one by one plus cos and yes so reduce everything to sin and cos so one plus sin square theta by cos square theta plus sin theta by cos theta divided by sin theta by cos theta sorry for the lack of space i'm writing here writing here okay so what will be the denominator like numerator will be cos square theta plus sin square theta so or else we can directly one plus tan square theta is equal to secant square theta anything anything you can do and in the denominator cos square ania so this is cos square theta and into cos theta by sin theta this is what i am getting okay so you can just give me my divided by tan theta anything will do anything if you if you are able to solve please so whatever striking striking my head i am doing that so one by sin theta cos theta divided by cos square this is cos theta time sin theta and then this is what you wanted isn't it because this is nothing but secant theta plus cos cos sorry cos secant theta at times you might veer away no problem but then okay never mind so there could be multiple ways of solving again so there could be tan as Surya was mentioning tan for you straight away right sin by cos cos by sin and do simplifications and you will end up getting that okay fair enough but this deserves three months yes that also can help no problem you can do that aditya so you can start with sin and sin by cos and eventually get the RHS whichever works there will be multiple groups in identities next this one dakshan monish very good in these kind of question there is one thing you can do yes rn i'll explain i'll i'll i'll give you give all of you a hint it will become easier if you do this what take this entire thing on the left hand side you don't need to convert anything just take that part to the left hand side good srishti good adity very good so the moment you take it to the other side it will be very very simple no problem monish you see the thing is you don't start from here let's say you start from this this part that's what i'm saying so you don't start from whatever it's given so can i not start like this you can do that no problem at all what i'm doing is this i will show you deco uh cot minus cosecant no you don't need to deco what i'm saying is this is as but as proving sin theta by cot theta plus cosecant theta minus sin theta upon cot theta minus cosecant theta right is equal to two this is so this identity is now you can always define the identity like that so prove this no problem so what is this sin theta common so let's take sin theta common and then what you will get cot theta minus cosecant theta then minus cot theta and minus cosecant theta divided by cot square theta minus cosecant square theta okay and then what is it this is nothing but sin theta and here this cot theta cot theta item will go and this is minus two cosecant theta divided by minus one isn't it cot cot square theta minus cosecant is right and then hence it is two into sin theta into cosecant theta sir so we are allowed to transpose expression in proving question see in the initial you can redefine it that's not let's say you don't want to transpose it now you know how to once let's say if you want to do like that only so now you know how to get to this level and then you rearrange the term when you you know rain but in my opinion this is okay there's no problem both LHS and RHS to be one cross theta and equate both LHS that is what is you know are the in my opinion don't do that together don't do together LHS RHS together so what I did is I just read you know so this identity is this so I've just recreated the identity and I did not do together I started from LHS once and then ended up getting RHS in that case okay so you can do adopt this method no problem right or else else if you don't want to do that then what or you have to do this no no not or yes you can do from RHS as well but two is a problem here so what you otherwise you can do is you can start like this LHS sin theta by cot theta plus cosecant theta minus one you start like this okay minus one so is equal to and then you prove this to be equal to one plus sin theta by cot theta minus cosecant theta okay so what is this sin theta minus cot theta minus cosecant theta divided by cot theta plus cosecant theta is it now you want cot minus cosecant in the denominator is it so hence what will you do or will that help so if I get that and sin also outside hmm so one way otherwise is now what you need to do is once you have done that okay then you can write this as one huh then you write this as one plus sin theta minus cot theta minus cosecant theta by cot theta plus cosecant theta minus one can I write like that can I or not is it okay see I'm trying to you know slowly move towards my destination so hence if I prove somehow is that okay so far so good I just added one and minus one yes or no did you get the point guys I need a confirmation are you with me is it okay so I simply started with LHS subtracted one from it okay and then try to simplify and then I added one and minus one so why because this one is needed here so I took I split this two one was sorry hello saying something someone was saying something anyways now why am I not you know doing sin and cos again because I have to come back to cot and cosecant so hence I am getting some some kind of hints from here now can you see that let us now resolve this further so let me write it here what is this this is one plus sin minus cot minus cot theta and then what is it oh great people are getting disconnected or beach may joining eight minute so sin theta minus cot theta minus minus cosecant theta minus cot theta and minus cosecant theta is it divided by cot theta plus cosecant theta right now this cot and this cot will go okay so this cot and this cot will go so what is left is equal to one plus so just because you wanted to do from LHS to RHS we are doing this so one plus sin theta okay I want sin theta also so I'm taking one one plus sin theta and what is left inside first let us simplify this so this is minus two cosecant theta divided by cot theta plus cosecant theta so this will you know get stretched a bit so sin I can take common 2 by cos yeah so I just need some space actually move forward okay wait where do I go I don't have see okay anyways what I'm trying to say is here you can extract sin again so some space I will extract so one plus sin theta and then what is this one minus two upon five marker right see you never know you know there is so hence for the three marker part I have already showed you how to do it oh sorry correct but just in case you want you can do it you know you need some more space to solve that's it but I'm saying you can try this I this is very much doable so this is one by two sin square theta sorry I don't have space here wait and wait now let us finish it oh if I remove it everything will go oh okay it's there so what I was saying is this let me copy paste so where was I so don't lose patience we will solve it oh come here no this is creating trouble wait a minute guys wait wait wait okay so what I was saying is this that we had one minus yeah sorry I am able to write now so what was the thing thing was let me just copy so one plus sin minus cot minus cosecant so one plus sin theta minus cot theta minus cosecant theta divided by cot theta plus cosecant theta I hope this was correct this one then minus one also isn't it let's let's check once again sin theta plus sin theta by cot theta plus cosecant theta minus one sin minus cot minus cosecant divided by this so this is one plus and minus very good so now let's proceed from here so hence what do I get I get one plus sin theta minus cot theta minus cosecant theta minus cot theta minus cosecant theta so I'm taking lcm either and divided by cot theta plus cosecant theta that's what I wanted to do there okay so this cot and this cot will go and I am left with one plus and let the sin theta thing come out because I anyways need sin theta so this will be one minus two cosecant so yes sir so how are you cancelling cot theta I'm sorry this is minus oh sorry my bad my bad my bad here I thought it is plus thanks thanks for the this thing so it is not going to be calc cancel thank you so deleted I thought it was plus it was supposed to get subtracted but anyways so anyways so one plus one plus what am I doing I am now taking sin theta minus two so better we can take the plus one part it will be then cot will get cancelled better you take plus one but then I need plus one here no I can't take plus one root because one is one is sacrosanct I can't leave one this is the part of the question okay is it so this has to be there so understood why am I not touching this one and this one I'm taking this one because simply simply because I need that one anyways just bear with me for some time so two plus cosecant theta yeah okay so far so good am I right yep so far so good so hence I need what I need one more sign right so one plus sin theta I'm extracting now now tell me whether I'm right or not two so cot by sin so this will be cot theta by sin theta plus cosecant theta by sin theta divided by cot theta plus cosecant theta am I right guys is it making sense so see I'm I'm step by step I'm getting what I intend to right but now there is one more problem what is the problem I need minus over here so what is the problem let's introduce minus so cot theta minus cosecant theta say up divide so if you really want to do like that now I get this and here you multiply also cot theta minus and now what is our job our job is to show that this is one yes or no so that's if that is one can you just you can just separate the denominator in the previous step how oh so like you can take sin theta by cot theta plus cosecant theta minus two cot theta plus cosecant theta by cot theta plus cosecant theta you separate but then my problem is I need a negative sign in the denominator it is not the plus sign here no so see what is the what is the sign here negative oh yes sorry so we have to get a negative sign over there so all this tamasha is because of that right so why so now if you see we are getting the intended result one plus sin theta upon cot theta minus cosecant theta right one why one here because one one of the ones you have taken on the left hand side here you can see that one I hope this pointer is visible so now you have to just you know some simplify somehow and prove that this is actually or this entire thing the top one is nothing but cot theta plus cosecant theta so is that so is this entire thing this one the one which I'm circling is equal to cot theta plus cosecant theta if you somehow prove that that's done I think that will be there but I am leaving it here why because anyways let's not you know get into that the proof is there it is the other one will become little lengthy so hence you can always do this so simplify this has been purposefully given like that so hence you can you separate the two out of it or not was the question okay or if someone has done by sign and cot that's fine never mind let's complete our set of questions and you can always adopt this method no problem chalo do this yes I will try yeshika let's finish it and toward the end we get some time we will come back to the problem but you know it can be solved that that was as well no problem do this one then anish says done this is a fairly simple question I believe oh yeah you have to club sign plus cost together I believe and then it should be done so just club this together so yeshika dance ray has done aren't done moniesh done so club these two together very good good to see lots of people are participating and responding actively very good guys again keep that goal in your site what is the goal what is the goal what's the goal what's the goal goal has to be in site always right very good keep the goal in mind now so hence what did I say just club them together so sine theta plus cos theta plus one and sine theta plus cos theta minus one you can sense this is a square minus b square form and there is secant theta into cosecant theta as well so this is nothing but sine theta plus cos theta whole squared minus one correct and this is secant theta times cosecant theta okay so expand this so this is nothing but sine square theta plus cos square theta plus two sine theta cos theta minus one times secant theta you can directly write this as well just for the sake of clarity okay yep so hence so this is this minus one that is zero so this is nothing but two times sine theta cos theta by sine theta cos theta is equal to two right so this is one so you have to write since okay yeshika what are you saying sir take the last part the other side sir if you take the secant theta and cosecant theta that side then you'll get these two multiplied is equal to two sine theta cos theta I don't need to you know again we are following that but let it be as far as possible from one side so I don't need to take them actually so you get you know get cancelled in this left hand side itself okay okay okay chalo next one oh again radical sine easy one you now know what to do come on come on jaha the moment you saw any radical sign the first thing which is should strike you is to come you know make it a perfect square somehow aditi is aditi has done okay aditi monish is also done okay very good anish done monish anish pa a little bit names aditi done aryan done so beta are done from surya p to everyone sir is it possible to solve by taking sine theta and one plus minus cos theta in the previous I'm sorry previous question okay we'll come back to previous kurya just hold on monish done swijjani done purav done akshita goel done akshita saying goen purav yeshika anike swisti satyam done done done done ananya apranav surya p akshita sir typo dot done sry comma typo dot done that's akshita okay uh yes others those two friends brothers and sisters of india what happened come on everyone's done done very good next come on come on come on come on now what to do here again so simply prisham is done very good prisham so second theta minus one divided by secant theta plus one so what I'll do is I will multiply this with secant theta minus one and root go right plus if you plus it you'll get secant theta plus one times secant theta plus one divided by secant theta minus one secant theta plus one very good that's what they're asking so hence nothing but uparwala baharagya secant theta minus one divided by root over secant square theta minus one so we'll go one more step let's not rush so secant theta plus one divided by root over secant square theta minus one again isn't it okay my dear friends so this is nothing but uh secant theta minus one so uh in the denominator you will get secant theta minus one divided by um tan theta plus secant theta plus one divided by tan theta which is equal to so two times secant theta upon tan theta which is equal to two times you can write if you wish cos theta into sin theta by cos theta which is equal to two times go secant theta here just write that secant square theta minus one is perfect no problem akshara is also done so far so good guys any trouble next now this is again a different way of asking about identities so secant theta plus tan theta is m show that this this again is very easy i mean yes this is a four mark sir which one is four mark monish this one so i've removed any you know i've removed the possibility of all the you know so even if now you previously it was four mark now it will be either three marker or two marker no four marker in the upcoming exam simply substitute and find that's it arian tendon done surya p done slash monish done guys are you writing down on a piece of paper or you're just doing mentally mind may subcoach very good please write yeshika has done very good prana of vj done substitute substitute done another question in which there is secant theta minus tan theta is equal to n yeah so there will be multiple varieties monish okay folks shall i try or you are going to finish this up quickly shri jenny says done ananya says done akshita says done adi bhijjad says done arun that he says done very good so let me now do it so hence i will do is this what i'm going to do is m co just substitute secant theta plus tan theta whole square plus one expand secant square theta plus tan square theta plus two secant theta and theta minus one divided by secant square theta plus tan square theta plus two secant theta tan theta plus one correct now now we will use the identity so secant secant square theta minus one co club cargo so what will this be secant square theta minus one plus tan square theta plus two secant theta tan theta and denominator may club these two so you'll get secant square theta plus tan square theta plus one show it like that and two secant theta tan theta okay now here is the beauty secant square theta is minus one is tan square theta plus tan square theta plus two tan theta secant theta divided by secant square theta plus secant square theta plus two tan theta secant theta okay now what you can take common what can we even take in common in the numerator so you can take two tan theta common within brackets it is tan theta plus secant theta in the denominator you can take two secant theta common and secant theta plus tan theta okay folks so this goes so two and two goes so tan theta by this thing is signed by cos into secant theta which is sign done okay and the i'm not writing the reasonings reasonings you can please mention the reasons there hanji do this one oh this is similar we did this did we do this there was a question similar to this but not exactly this one but same only yes similar only so that time it was one upon cos plus secant cos secant plus i think something that anyway start or hold on hold on this is similar try hard try this one leave that try this one done aniket says done done bad writing yeshika also says done good yeshika keep it up shurya done four people done writing steps is so annoying giving marks without seeing the steps is so annoying aniket ji dupta ji done done okay so please write steps if there are no steps how will you climb from ground floor to top floor now will you get down also you'll hurt yourself steps are important in life lift good one if examiner could understand our handwriting everything would be easy if examiner you know if if he has he gets a power of understanding handwriting why can't he directly get the power of read your minds katham enough so i can read munish's mind yes he is doing sin theta plus cos theta there okay he's doing right full marks doing exam connectors are literally like robots oh my god you have different you know so so much of this thing you have to revere them till they till they give you good marks at least okay so don't send wrong vibes okay they are listening anyways chalo so cos theta plus sin theta is root 2 cos theta show that cos theta what did you do guys i would have done this cos theta plus sin theta is equal to root 2 cos theta that means i can say root 2 times cos theta sorry squaring both sides and all not needed i will do like this what will i do i will write cos theta common root 2 minus 1 is equal to sin theta okay now i will say let cos theta minus sin theta is equal to x okay and what is given cos theta plus sin theta is equal to root 2 cos theta add both of them so you'll get 2 cos theta is equal to root 2 cos theta plus x so x is equal to 2 cos theta minus root 2 cos theta root 2 cos theta common you will get root 2 minus 1 is it it yeah and root 2 minus 1 cos theta is sin theta so x is root 2 sin theta done so this is since since it is what since we know that root 2 minus 1 cos theta is equal to sin theta okay so this is also you know you can do either ways which you want okay folks so this is done anyone any difficulty here no doubt shall we proceed this is mechanical very unlikely stress very unlikely that you will get an identity for pie marks but because then where will the heights and distance application problem go can you solve the last one by squaring both sides yes please do no problem you can square and do no problem negna to to answer your query which one oh you want to so you want me to show you that so those who are solving this question please solve let me solve that question back after this one by squaring so let me solve it here itself in the space so that question was i believe cos theta for sin theta so square both sides you'll get cos square theta plus sin square theta plus 2 cos theta sin theta 2 cos square theta correct yes so now what will you get you will get this implies cos square theta minus sin square theta minus 2 sin theta cos theta so you take everything on one side is equal to zero am i right cos either cos minus sin yes now do what this implies cos square theta plus sin square theta add sin square theta to it minus 2 sin theta cos theta and then on the other side you will get 2 sin square theta see so add 2 sin square theta sir one second shouldn't that be cos square plus sin square no this is 2 cos square no so 2 minus this cos on the other side i'm writing okay okay all right all right right so hence 2 cos square theta minus cos square theta will be cos square theta then this sign on the other side will become minus sin square and this plus 2 cos theta sin theta on the other side will become minus 2 sin theta cos theta and entire thing is zero now add 2 sin square theta to both sides why because i want to complete this square so now you'll get the desired result what is that so this is cos theta minus sin theta whole square is equal to 2 sin square theta and hence it will follow understood yes sir got it hanji this one done isn't this five marks identity sum no there is no five marks identity no they will not give you but practice karlona what is the problem okay so what is this this is a mechanical problem so and there is no brainer in this simply keep doing what multiplying am i right or or or or or second cube minus cosecant cube second cos cube minus sine cube okay so here what will i do i will convert everything into sine and cos they go one plus and i'm writing only in c and s okay so c by s plus s by c so you don't do this in the exam this is just to save space here hence okay this is s minus c divided by secant is 1 by c cube minus 1 by s cube that's it so this is take the common denominator s c this will be sc plus c square plus s square in 1 here s minus c and here it is going to be s cube minus c cube into c cube into s cube i hope all of you are getting it so i'm just writing in shorthand i don't have space hence otherwise please do not write in s and c and all that or if you are very you know you can start only with let theta be s let sine theta but that is please don't do that you never know people would not be happy okay anyways so what do i do now so it is sc plus c square plus s square i can directly write 1 then s minus c and then c cube s cube actually i will just you know i will make it c square s square divided by s cube minus this thing is s minus c s square plus c square plus sc so again this this will go this will this will go so c square s square okay clear no problem you should go now clear everyone i just did in very shorthand so you have to write all that sine cos and all any step you didn't understand see again i just cancelled here this step why because wait where did the pointer go yeah so this is 1 and this is sc so this this and this will go any problem guys hello people people problem no problem people so okay this one is easy do this one could you go back why not here this one aditya any doubt anywhere please ask okay do this one is transposition allowed you can or without that also you can do every time you don't need to start from LHS and go to RHS you can start with a standard identity so here you can start with an identity called cosecant square minus quad square theta is one this is that manifestation which we had discussed here that you know in the previous slide so you don't need to even start from the LHS what i'm saying is here you start with this cosecant square theta minus quad square theta is equal to one no problem no yes so you have cosecant theta minus cot theta is equal to one upon what is it um cosecant theta plus cot theta hmm yes or no so this is this one you can do now and what what else so that can be written as cosecant theta minus cot theta yeah you can either start from here and then do what or if i prove this now let us say one plus cosecant theta plus cot theta plus one upon cosecant theta minus cot theta okay and also you can you can see this also that one upon cosecant theta minus cot theta is going to be how much cosecant theta plus cot theta yes or no both are same right this and this both are same so hence start from here and then do what one plus cosecant plus cot can be written as cosecant theta minus cot theta plus one by cosecant theta minus cot theta can be written as cosecant theta plus cot theta hence it is two cosecant theta that is equal to two by sine theta okay hence you can write one upon cosecant theta plus cot theta minus one upon sine theta is equal to one by sine theta minus one upon cosecant theta minus cot theta proved so i didn't do any transposition nothing else what did i do i first proved that this is equal to this and i also proved that this is equal to this and then i started with adding these two quantity i got two upon sine theta and then i got that okay anyone has any other step or any other method in this someone did something else previous one which one aditya this one confused okay people are getting confused okay okay both questions i will repeat so here let me delete this step is clear is this clear whatever is there there on the board so far yes aditya is that okay okay so now after that what did i do s cube minus c cube so let the numerator be as it is s minus c c cube s cube rationalizing in the three mark question didn't get any other word for no problem yeah so here just a minute munish now what do you do s just check here this sc i had i had you know stuck it down and cancelled this and it has become two see there was a power three here power three here but i cancelled with this sc here there was a sc in the denominator aditya is that clear yep so that is clear then what is left this is s cube minus c cube what is the expansion s minus c s square plus c square plus sc no now what is s square plus c square one so one plus sc and this this will go no this will go and this s minus c s minus c will go so what is left c square s square clear now yep next this one so akshita what was the issue they go once again what did i do okay i did this i said i said that cosecant square theta minus cot square theta is one hence cosecant theta minus cot theta is equal to one upon cosecant theta plus cot theta is this clear similarly cosecant theta plus cot theta will be equal to one upon cosecant theta minus cot theta is this clear same thing now add these two add these two what will you get in the LHS if i write this i can write one plus cosecant theta plus cot theta plus one upon cosecant theta minus cot theta is equal to two cosecant theta if i add the LHS you'll get two cosecant theta which is nothing but two upon sine theta right now this two upon sine theta can be written as one by sine theta plus one by sine theta now you take one sine theta on the other side then let me one one by sine theta on one side right so let's take let's take this here and let's take this part there so what will you get one upon cosecant theta plus cot theta minus one upon sine theta is equal to one upon sine theta minus one upon cosecant theta minus cot theta now understood this is what was requirement of the question yep they go right now yes exactly same so using the manifestations which we talked about you can prove this so this question now anyway we are short of time so i'm just leaving it as you know a practice question it is easy you can do at home as well this is how it should be written they go wherever required reason is given a reason right so always i told you keep some 15 20 minutes towards the end of the exam and all the identity hardly there will be one or at max two questions right so when you come back write all the appropriate reasons wherever required and one column one by one step by step make it as clear as possible do not jump steps and hence done right lh is equal to rhs and this is what you have to do in identity see this is the question which was asked in the board exam which we didn't do but it's there okay in the in the slides you can solve them fair enough understood any problem below could squeeze at least to anyone if yes then do discuss with me i am available on phone or whatsapp please reach out to me and now we are done with trigonometry so 12 marks are in your pocket right before the exam you can try this in the upcoming pre-board so just go through the slides whatever we have discussed and there is no reason you will not get 12-1-12 at least in trigonometry is that clear okay will the ppt already shared oh you please be informed all the all the ppt's are shared in the same where all till now whatever we have conducted all the ppt's have been shared in pdf form no no it's there and now we are here look in your lernist go to comprehensive revision program here all the ppt's are attached to the same life-class link so you can see there are four attachments are there attachment attachment attachment everywhere attachment is there so this is mathematical quadratic equation this is electricity this is mathematics then week two yesterday there was acid base that is there now today after this class i will attach my ppt here as well okay so use it all everything is given which is required for acquiring sentiment in maths the ppt's should be good enough okay so we have seen that in the past you can use them you can try this this exam itself upcoming pt fair enough i hope the session was useful to all of you you learned something it was useful and again reiterating what's our target target is to get 100 marks sentiment in all all subjects as much as possible right it's very much possible you are on right track just be cool and calm and composed do not panic you have already done so much of work you have written so many exams so there is no reason why you should not be getting sent in fair enough with all those words let me sign off i think we are going to meet next year only because on 31st there is no no class so we will be meeting you on i think second of tomorrow there is a bio class then there is no class this year we will i will meet you next year only so wishing you everyone wishing everyone here a very very happy new year corona free year great and successful milestone walla year all of you get lots of sentence so that you can cherish that throughout your year my best wishes to all of you bye bye folks enjoy and see you again next year happy new year happy new year to everyone


Thank you very much. All right. Here we go now. It's time for really nerdy stuff So yeah, this came about because I actually heard reporters saying to me after docker became really you know You know all the hype cycle came up people are saying oh, we don't need the operating system anymore and I've heard this time and time again and as a former Technology journalist I can tell you that this is part of the normal hype cycle and this is my attempt to say whoa Hold on ladies and gentlemen. That's not quite all The story so here we go a brief history of operating systems in the beginning hydrogen helium Stars dead stars more stars heavier elements Silicon iron copper, you know the drill they all come together Some carbon gets together it comes up that seemed like a good idea of life You know whatever and then they got together Let's take all those rocks put them together and make tools out of them And one of those tools is the computer Okay, but then they said hey we need to write some abstract math to sort of talk to these little pieces of Silicon and metal to actually make them do something and those would be the operating systems And we have so many operating systems right now. It's not even funny We have this this has an operating system that camera has an operating system that has enough That whatever that is on the wall that has an operating system the projector has an operating It's all operating system. They're everywhere. We live in a very operating centric Operating system centric world now. We've improved on this a little bit. We've said to ourselves, okay It used to be you know if you have a piece of metal, you're gonna have an operating system on it Unless it's a Windows machine, and we're not going to call it OS because well, no Yeah, ladies and gentlemen cheap shot number one so But virtualization helped this we abstracted the operating system away from the metal We could say okay instead of one OS per piece of Hardware we could actually have multiple OSes running and that's what virtualization helped us do But we still have the same problems No matter where you go with these operating systems You're still having to deal with the inherent problems and an operating system has especially if you're a developer or a Sysadmin these are still issues that have to be contended with we still live in an OS centric world You know we do well on some things on the planet earth We currently hold all the mr. And this universe titles in the entire universe. I'm just saying so we're doing really well there but It's still been an operating system centric world and we're sort of now moving away from that as soon as we did People are like oh, maybe operating systems are dead.  You're putting it on an operating system Oh the joys of finding out that the latest security patch to that operating system just broke your app You know that you've worked so diligently to do these are real You know this happens all the time admins have the same issues They have to deal with the developers complaining about those security patches Plus they have to get those security patches out because you know 500 people from some country across the planet are trying to get into that computer as we speak Okay, and they have to deal with the inherent flaws devops really doesn't do anything better I mean it makes the the time to you know app development to deployment better But still the underlying issue with dealing with operating systems is Still a factor here. You can streamline the process you can streamline it with devops deployments you can streamline it with virtualization and cloud, but these are still going to have the same problems The virtualization and cloud they're the first steps You can make things better You can treat instead of treating operating systems like pets You can treat them like cattle or if you're a vegan instead of treating them like heirloom tomatoes You can treat them like a field of wheat Okay, I just want to make sure we get that all in there. Okay, but now now And I mentioned this earlier we have containers now containers are not new They've been around for a while BSD calls them jails. So layers call some zones Linux They're built into the kernel. They're called Linux containers Alex C But now they're getting a lot more hype because of things like Docker Containers which are if you're not familiar they are basically file system and namespace containers of Libraries and applications that run just what you need to run the application They're very app-centric if you write an app Put it inside a container give it only what it needs to run You can take that app especially with Docker pick it up move it and drop it on another Docker machine done You don't care what the operating system is doing underneath As long as it's running Docker, you're fine because Docker will take care of everything else that layer Helps you make an app-centric world So for that reason containers are pretty much awesome But not all the way Containers are new. I mean the old container technology is Relatively mature, but the way we're using them now and we're trying to do things with containers that haven't been done before So the technology is sort of growing and stretching. We see growing pains containers are leaky We have containers sprawl where people just say I'm gonna containerize everything and woohoo And now you've got a million containers and nowhere to put them. So there are still some inherent issues there So I've done a pretty good job now slamming operating systems and a fairly decent job talking up containers So at this point you might be asking yourself. Why does anybody care about operating systems anymore? Now this guy he may care about operating system a little too much It's okay to love your operating system Okay, but seriously, why do we care? Why do we need to have operating systems? Is this model really? Antiquitated are we really moving to something else and I would say to you No, we're not and you'll probably be suspicious because it's like hmm. He works for Red Hat and Overt is a virtual machine manager that deals with virtual machines that still depend on operating systems Why should I listen in the sky? Well because All of the developments that I told you about Still were built on top of operating systems so containers and virtual machine managers and hypervisors and all the other innovations that we've seen Since the dawn of the competing age Since we figured out how to get a bunch of rocks and metal and silicon together and actually Organized them to do really kind of cool things for us Like Twitter, that's cool Okay, that's all coming from operating systems This is the model that we've started with and for now This is the model that I believe that we will continue to progress with and innovate on To give you an example. What why I feel this way look at micro manufacturing or 3d printing. Oh My goodness. I've lost Okay 3d printing I've totally lost this game 3d printing. It's really cool.  I'm gonna beat on myself So every journey that we make with operating system everything you do with an application Whether short or long this is the Indian Mars Explorer that was recently got to Mars from India You know big achievement for that company it all started from a small place like this Small vocational high school in India. These are all this we all start from the same place the same Deep background IT has to be innovated somewhere and It's going to be on the operating system moving forward They are the foundation like it or not Don't believe all the hype containers are great tools But they're not going to be to be all-in-all for now You're still going to need to work on operating systems and still innovate So with that and my overly long lightning talk, which is now a complete thunderstorm. Don't be a hater