L.H.S. $=4\left[\cot ^{-1}(3)+\operatorname{cosec}^{-1} \sqrt{5}\right]$
$=4\left[\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{2}\right)\right]$
$=4\left[\tan ^{-1}\left(\frac{\frac{1}{3}+\frac{1}{2}}{1-\frac{1}{3} \times \frac{1}{2}}\right)\right]=4 \tan ^{-1}\left[\frac{5}{6} \times \frac{6}{5}\right]$
$=4 \tan ^{-1}(1)=4 \times \frac{\pi}{4}=\pi=\text { R.H.S. }$
