Question

The value of ${\sin }^{-1}\left[\cos \{{\cos }^{-1}\left(\cos x\right)+{\sin }^{-1}\left(\sin x\right)\}\right]$, where $x\in (\frac{\pi }{2},\pi )$ is equal to

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

Answer 1

The correct option is D $-\frac{\pi }{2}$

The value of ${\sin }^{-1}\left[\cos \left\{{\cos }^{-1}\left(\cos x\right)+{\sin }^{-1}\left(\sin x\right)\right\}\right]$

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

$=-\frac{\pi }{2}$