Question

If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.

Answer 1

$$\begin{gathered} \tan \left(2A\right)=\tan (A+A) \\ =\tan (A+B+A-B) \\ =\frac{\tan (A+B)+\tan (A-B)}{1-\tan (A+B)\tan (A-B)} \\ =\frac{x+y}{1-xy} \end{gathered}$$

$$\begin{gathered} \tan 2B=\tan \left(B+B\right) \\ =\tan \left(B+A+B-A\right) \\ =\frac{\tan \left(A+B\right)+\tan \left(B-A\right)}{1-\tan \left(A+B\right)\tan \left(B-A\right)} \\ =\frac{\tan \left(A+B\right)-\tan \left(A-B\right)}{1+\tan \left(A+B\right)\tan \left(A-B\right)}\left[\tan \left(-\theta \right)=-\tan \theta \right] \\ =\frac{x-y}{1+xy} \end{gathered}$$