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Question
Write the value of \[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right)\]
Solution
\[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right) = \tan\left[ \tan^{- 1} \frac{2 \times \frac{1}{5}}{1 - \left( \frac{1}{5} \right)^2} \right]\]
\[ = \tan\left( \tan^{- 1} \frac{\frac{2}{5}}{\frac{24}{25}} \right)\]
\[ = \tan\left( \tan^{- 1} \frac{5}{12} \right)\]
\[ = \frac{5}{12}\]
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