If tan (A + B) = p, tan (A – B) = q, then the value of tan 2A in terms of p and q is ___________.

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Solution

Given (A + B) = p
tan (A – B) = q

$\begin{array}{rcl} \tan 2 A & = & \tan [A + A] \\ = & \tan [(A + B) + (A - B)] \\ = & \frac{\tan (A + B) + \tan (A - B)}{1 - \tan (A + B) \tan (A - B)} [using identity \tan (x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}] \\ \tan 2 A & = & \frac{p + q}{1 - p q} \end{array}$

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