Write the value

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

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

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

$=\frac{5}{12}$