If $tan (A + B) = x$ and $t a n (A - B) = y$ find the value of $tan 2 A$ and $tan 2 B$

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Solution

The number $2 A$ can be written as $((A + B) + (A - B))$ .
$tan 2 A = tan ((A + B) + (A - B))$
$= \frac{tan (A + B) + tan (A - B)}{1 - tan (A + B) tan (A - B)}$
$= \frac{x + y}{1 - x y}$
The number $2 B$ can be written as $((A + B) - (A - B))$ .
$tan 2 B = tan ((A + B) - (A - B))$
$= \frac{tan (A + B) - tan (A - B)}{1 + tan (A + B) tan (A - B)}$
$= \frac{x - y}{1 + x y}$

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