SOLUTION : $2 \tan ^{-1} \frac{1}{5}=\tan ^{-1}\left(\frac{\frac{2}{5}}{1-\frac{1}{25}}\right)=\tan ^{-1}\left(\frac{\frac{2}{5}}{\frac{24}{25}}\right)=\tan ^{-1} \frac{5}{12} \\$
$=\tan \left(\tan ^{-1} \frac{5}{12}+\frac{\pi}{4}\right)=\frac{\tan \tan ^{-1} \frac{5}{12}+\tan \frac{\pi}{4}}{1-\left(\tan \tan ^{-1} \frac{5}{12}\right) \tan \frac{\pi}{4}}=\frac{\frac{5}{12}+1}{1-\frac{5}{12}}=\frac{\frac{17}{12}}{\frac{7}{12}}=\frac{17}{7}$
