If tan(A+B)=x and tan(A-B) =y, find the values of tan2A and tan2B.

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Solution

We have,
tan(A+B)=x and tan(A-B) =y
Now, tan2A= tan[(A+B)+(A-B)]
$= \frac{t a n (A + B) + t a n (A - B)}{1 - t a n (A + B) \times t a n (A - B)}$
$= \frac{x + y}{1 - x y}$
$∴ t a n 2 A = \frac{x + y}{1 - x y}$
Now, tan2B= tan[(A+B)-(A-B)]
$= \frac{t a n (A + B) - t a n (A - B)}{1 + t a n (A + B) \times t a n (A - B)}$
$= \frac{x - y}{1 + x y}$
$∴ t a n 2 B = \frac{x - y}{1 + x y}$

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