If tan (A + B) = x and tan (A − B) = y,
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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