# is equal to:

Question:

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

1. (1) $\frac{75}{56}$

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

3. (3) $\frac{56}{33}$

4. (4) $\frac{65}{33}$

Correct Option: 2,

Solution:

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

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

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

$=\operatorname{cosec}\left(\tan ^{-1}\left(\frac{5}{12}\right)+\cos ^{-1}\left(\frac{4}{5}\right)\right)$

Let $\tan ^{-1}(5 / 12)=\theta \Rightarrow \sin \theta=\frac{5}{13}, \cos \theta=\frac{12}{13}$

and $\cos ^{-1}\left(\frac{4}{5}\right)=\phi \Rightarrow \cos \phi=\frac{4}{5}$ and $\sin \phi=\frac{3}{5}$

$=\operatorname{cosec}(\theta+\phi)$

$=\frac{1}{\sin \theta \cos \phi+\cos \theta \sin \phi}$

$=\frac{1}{\frac{5}{13} \cdot \frac{4}{5}+\frac{12}{13} \cdot \frac{3}{5}}=\frac{65}{56}$