Question

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

Answer 1

$\tan \left[2{\tan }^{-1}\frac{1}{5}\right]$

We know that $2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$

Here, $x=\frac{1}{5}$

$=\tan \left[{\tan }^{-1}\left(\frac{2\times \frac{1}{5}}{1-(\frac{1}{5}{)}^2}\right)\right]$

$=\tan \left[{\tan }^{-1}\left(\frac{2\times \frac{1}{5}}{1-\frac{1}{25}}\right)\right]$

$=\tan \left[{\tan }^{-1}\left(\frac{\frac{2}{5}}{\frac{24}{25}}\right)\right]$

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

$=\tan \left[{\tan }^{-1}\left(\frac{5}{12}\right)\right]$

$=\frac{5}{12}$