The value of $\mathrm{cosec}\left(2{\cot }^{-1}5+{\cos }^{-1}\frac{4}{5}\right)$ is

$\mathrm{cosec}\left(2{\cot }^{-1}5+{\cos }^{-1}\frac{4}{5}\right)$

$=\mathrm{cosec}\left(2{\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{3}{4}\right)$

$\begin{array}{l}=\mathrm{cosec}\left({\tan }^{-1}\left(\frac{2\times \frac{1}{5}}{1-\frac{1}{25}}\right)+{\tan }^{-1}\frac{3}{4}\right)\\ =\mathrm{cosec}\left({\tan }^{-1}\frac{5}{12}+{\tan }^{-1}\frac{3}{4}\right)=\mathrm{cosec}\left\{{\tan }^{-1}\left(\frac{\frac{5}{12}+\frac{3}{4}}{1-\frac{5}{12}\times \frac{3}{4}}\right)\right\}\\ =\mathrm{cosec}\left({\tan }^{-1}\frac{56}{33}\right)=\mathrm{cosec}\left({\mathrm{cosec}}^{-1}\frac{65}{56}\right)=\frac{65}{56}\end{array}$