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Question
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
Solution
\[\tan(2A) = \tan(A + A)\]
\[ = \tan(A + B + A - B)\]
\[ = \frac{\tan(A + B) + \tan(A - B)}{1 - \tan(A + B)\tan(A - B)}\]
\[ = \frac{x + y}{1 - xy}\]
\[\tan 2B = \tan \left( B + B \right)\]
\[ = \tan \left( B + A + B - A \right)\]
\[ = \frac{\tan \left( A + B \right) + \tan \left( B - A \right)}{1 - \tan\left( A + B \right)\tan\left( B - A \right)}\]
\[ = \frac{\tan\left( A + B \right) - \tan\left( A - B \right)}{1 + \tan\left( A + B \right)\tan\left( A - B \right)} \left[ \tan\left( - \theta \right) = - \tan \theta \right]\]
\[ = \frac{x - y}{1 + xy}\]
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