Question

Write the value of $\tan \left(2{\tan }^{-1}\frac{1}{5}\right)$

Answer 1

$$\begin{gathered} \tan \left(2{\tan }^{-1}\frac{1}{5}\right)=\tan \left[{\tan }^{-1}\frac{2\times {\displaystyle \frac{1}{5}}}{1-{\left({\displaystyle \frac{1}{5}}\right)}^2}\right] \\ =\tan \left({\tan }^{-1}\frac{{\displaystyle \frac{2}{5}}}{{\displaystyle \frac{24}{25}}}\right) \\ =\tan \left({\tan }^{-1}\frac{{\displaystyle 5}}{{\displaystyle 12}}\right) \\ =\frac{{\displaystyle 5}}{{\displaystyle 12}} \end{gathered}$$