The value of tan$\left[2{\tan }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle 5}}\right)-\frac{{\displaystyle \pi }}{{\displaystyle 4}}\right]$  is 

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

$\tan \frac{\alpha }{2}=\frac{1}{5}$

$\Rightarrow \tan \alpha =\frac{2\tan \frac{\alpha }{2}}{1-{\tan }^2\frac{\alpha }{2}}$

$=\frac{\frac{2}{5}}{1-{\tan }^2\frac{\alpha }{2}}$

$=\frac{\frac{2}{5}}{1-\frac{1}{25}}$

$=\frac{2}{5}\times \frac{25}{24}$

$=\frac{5}{12}$

$\tan \left(\alpha -\frac{{\displaystyle \pi }}{{\displaystyle 4}}\right)=\frac{\tan \alpha -\tan \frac{\pi }{4}}{1+\tan \alpha .\tan \frac{\pi }{4}}$

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