Take the 2-minute tour × TrigExpand@Tan[x + y]   $\frac{\sin (x)
\cos (y)}{\cos (x) \cos (y)-\sin (x) \sin (y)}+\frac{\cos (x)
\sin(y)}{\cos (x) \cos (y)-\sin (x) \sin (y)}$  but I expected
Evaluating TrigExpand with arguments $\tan (x-y)$ or $\tan
(3*x)$ also returns results in terms of sine and cosine. What
should I do to get results in terms of the tangent?
share|improve this question add comment  2 Answers  up vote 10
down vote accepted  One way to induce Mathematica to simplify to
Tan functions is to introduce the arguments as inverse tangents,
as in $x\equiv \arctan a$ and $y\equiv \arctan b$. Then you
could write for example    TrigExpand@Tan[ArcTan[a] +
ArcTan[b]]] /. {a -> Tan[x], b -> Tan[y]}  (* ==> (Tan[x] +
Tan[y])/(1 - Tan[x] Tan[y]) *)  or more generally with an
expression containing x and y:  expr = Tan[3 x];     expr /. {x
-> ArcTan[a], y -> ArcTan[b]}]] /. {a -> Tan[x],    b -> Tan[y]}
(* ==> (Tan[x] (-3 + Tan[x]^2))/(-1 + 3 Tan[x]^2) *)   Referring
to the answer by @belisarius, I though