Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

Given, $cosec\left[2{\tan }^{-1}\left(\frac{1}{5}\right)+{\tan }^{-1}\left(\frac{3}{4}\right)\right]$

$=cosec\left[{\tan }^{-1}\left(\frac{2\times \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}{1-\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\right)}\right)+{\tan }^{-1}\left(\frac{3}{4}\right)\right]$

$=cosec\left[{\tan }^{-1}\left(\frac{5}{12}\right)+{\tan }^{-1}\left(\frac{3}{4}\right)\right]$

$=cosec\left[{\tan }^{-1}\left(\frac{\frac{5}{12}+\frac{3}{4}}{1-\frac{5}{12}\times \frac{3}{4}}\right)\right]$

$=cosec\left[{\tan }^{-1}\left(\frac{56}{33}\right)\right]=\frac{65}{56}$

option $\left(1\right)$