The numerical value of \(\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)\)

\(\text {Let } 2 \tan ^{-1}\left(\frac{1}{5}\right)=x \)
\( \therefore \tan ^{-1}\left(\frac{1}{5}\right)=\frac{x}{2} \)
\( \therefore \tan \frac{x}{2}=\frac{1}{5} \)
\( \text { Using } \)
\( \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^2 \theta} \)
\( \therefore \tan x=\frac{2 \tan ^{\frac{x}{2}}}{1-\tan ^2 \frac{x}{2}}\)
\(\Rightarrow \tan x=\frac{2 \times \frac{1}{5}}{1-\frac{1}{25}} \)
\( \Rightarrow \tan x=\frac{5}{12}...(i)\)
\(\text {Consider } \tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right) \)
\( =\tan \left(x+\frac{\pi}{4}\right) \)
\( =\frac{\tan x+\tan \frac{\pi}{4}}{1-\tan x \cdot \tan \frac{\pi}{4}} \)
\( =\frac{\frac{5}{12}+1}{1-\frac{5}{12}} \)
\( =\frac{\frac{17}{12}}{\frac{7}{12}}=\frac{17}{7}\)