Prove the following
Question:

$\operatorname{cosec}\left[2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :

1. $\frac{56}{33}$

2. $\frac{65}{56}$

3. $\frac{65}{33}$

4. $\frac{75}{56}$

Correct Option: , 2

Solution:

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

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

$=\operatorname{cosec}\left[\tan ^{-1}\left(\frac{56}{33}\right)\right]=\frac{65}{56}$ option $(2)$