Question:
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
Solution:
$\tan (2 A)=\tan (A+A)$
$=\tan (A+B+A-B)$
$=\frac{\tan (A+B)+\tan (A-B)}{1-\tan (A+B) \tan (A-B)}$
$=\frac{x+y}{1-x y}$
$\tan 2 B=\tan (B+B)$
$=\tan (B+A+B-A)$
$=\frac{\tan (A+B)+\tan (B-A)}{1-\tan (A+B) \tan (B-A)}$
$=\frac{\tan (A+B)-\tan (A-B)}{1+\tan (A+B) \tan (A-B)} \quad[\tan (-\theta)=-\tan \theta]$
$=\frac{x-y}{1+x y}$
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