Question

If $x\in (\frac{3\pi }{2},2\pi )$, then the value of the expression ${\sin }^{-1}\left[\cos \{{\cos }^{-1}\left(\cos x\right)+{\sin }^{-1}\left(\sin x\right)\}\right]$

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

Answer 1

The correct option is B $\frac{\pi }{2}$

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