Trig has two identities called ratio identities. This label can be confusing, because all the trig functions are defined by ratios. Somewhere along the line, however, mathematicians thought this description was perfect for these two identities, because they’re basically fractions made up of two trig functions, one above the other, in each. The ratio identities create ways to write tangent and cotangent by using the other two basic functions, sine and cosine.

The ratio identities are


These two identities come from the simplification of a couple of complex fractions. If you use the basic definitions for sine, cosine, and tangent,


then you can see that


Likewise, because cotangent is the reciprocal of tangent,