The value of $\cot \left(\frac{\pi }{4}-2{\cot }^{-1}3\right),$ is 

    $\cot \left(\frac{\pi }{4}-2{\cot }^{-1}3\right)$

$=\frac{1}{\sin \left(\frac{\pi }{4}-2{\tan }^{-1}\frac{1}{3}\right)}=\frac{1+\tan \left(2{\tan }^{-1}\frac{1}{3}\right)}{1-\tan \left(2{\tan }^{-1}\frac{1}{3}\right)}$

$=\frac{1-\tan \left({\tan }^{-1}\frac{3}{4}\right)}{1-\tan \left({\tan }^{-1}\frac{3}{4}\right)}=7\left[\because 2{\tan }^{-1}\frac{1}{3}={\tan }^{-1}\left(\frac{2/3}{1-\frac{1}{9}}\right)={\tan }^{-1}\frac{3}{4}\right]$