Question

The value of $\sin^{-1} \cos (sin^{-1}x)+\cos^{-1} \sin(\cos^{-1}x)$ is

• A. 0
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{2}$
• D. $\pi$

Answer 1

Answer: (C)

$sin^{-1} \left[ cos\left(sin^{-1}x\right)\right] + cos^{-1}\left[sin\left(cos^{-1}x\right)\right] $
$= sin^{-1}\left[sin\left(\frac{\pi}{2} -sin^{-1}x\right)\right] + cos^{-1} \left[cos\left(\frac{\pi}{2}-cos^{-1}x\right)\right]$
$= \frac{\pi}{2} -sin^{-1} x + \frac{\pi}{2}-cos^{-1}x $
$ = \pi-\left(sin^{-1}x +cos^{-1}x\right) $
$ = \pi- \frac{\pi}{2} $
$= \frac{\pi}{2}$