The value of $tan \begin{Bmatrix} 2 tan^{-1} \left(\frac{1}{5}\right) + \frac{\pi}{4}\end{Bmatrix}$ is :

$\frac{17}{7}$

The correct answer is Option (3) → $\frac{17}{7}$

$\tan\left\{2 \tan^{-1} \left(\frac{1}{5}\right) + \frac{\pi}{4}\right\}$

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

$\tan^{-1}\left(\frac{10}{24}\right)=\tan^{-1}\frac{5}{12}$

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

$=\frac{\frac{5}{12}+1}{1-\frac{5}{12}}=\frac{17}{7}$