Question

If $\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}$ then a value of $x$ is-

(1) 1

(2) 3

(3) 4

(4) 5

Answer 1

SOLUTION: Since, $\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2} \Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)+\sin ^{-1}\left(\frac{4}{5}\right)=\frac{\pi}{2}$

$\Rightarrow  \sin ^{-1}\left(\frac{x}{5}\right)=\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{5}\right)  \sin ^{-1}\left(\frac{x}{5}\right)=\cos ^{-1}\left(\frac{4}{5}\right) \\$

$\Rightarrow \quad \sin ^{-1}\left(\frac{x}{5}\right)=\sin ^{-1}\left(\frac{3}{5}\right)  \Rightarrow  x=3$

Hence Option 2 is Correct.