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

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Solution

Given: $tan (A + B) = x$ and $tan (A - B) = y,$
We know,
$2 A = (A + B) + (A - B)$

$\Rightarrow tan 2 A = \frac{tan (A + B) + tan (A - B)}{1 - tan (A + B) - tan (A - B)}$

$\Rightarrow tan 2 A = \frac{x + y}{1 - x y}$

Also,

$2 B = (A + B) - (A - B)$

$\Rightarrow tan 2 B = \frac{tan (A + B) - tan (A - B)}{1 + tan (A + B) - tan (A - B)}$

$\Rightarrow tan 2 B = \frac{x - y}{1 + x y}$

Hence,

$tan 2 A = \frac{x + y}{1 - x y}$ , $tan 2 B = \frac{x - y}{1 + x y}$

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