# If $\sin ^{-1}\left ( \frac{x}{5} \right )+cosec^{-1}\left ( \frac{5}{4} \right )= \frac{\pi }{2},$ then the values of $x$ is Option 1) $4$ Option 2) $5$ Option 3) $1$ Option 4) $3$

P Prateek Shrivastava

As we learnt in

Important Results of Inverse Trigonometric Functions -

$\sin ^{-1}x + \cos ^{-1}x = \frac{\pi }{2}$

- wherein

When $\left | x \right |\leqslant 1$

$\sin ^{-1}\frac{x}{5}+\cos ^{-1}\frac{5}{4} = \frac{\pi }{2}$

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

taking cos on both sides,

$\cos \sec^{-1}\frac{5}{4}=\frac{x}{5}$      $\Rightarrow x=5\ cos\ cosec^{-1}\left(\frac{5}{4} \right )=5\times \sqrt{\left(\frac{5^{2}-4^{2}}{5} \right )}=3$

$x=3$

Option 1)

$4$

Incorrect

Option 2)

$5$

Incorrect

Option 3)

$1$

Incorrect

Option 4)

$3$

Correct

Exams
Articles
Questions