Question

Calculate the value of ${\sin }^{-1}\cos \left({\sin }^{-1}x\right)+{\cos }^{-1}\sin \left({\cos }^{-1}x\right)$. where $\left|x\right|\le 1$

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

Answer 1

The correct option is C $+\frac{\pi }{2}$

Given, ${\sin }^{-1}\cos \left({\sin }^{-1}x\right)+{\cos }^{-1}\sin \left({\cos }^{-1}x\right)$

$={\sin }^{-1}\cos \left({\cos }^{-1}\sqrt{1-x^2}\right)+{\cos }^{-1}\sin \left({\sin }^{-1}\sqrt{1-x^2}\right)$

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

Since ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}\forall |x|\le 1$