Question

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

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

Answer 1

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

$co{s}^{-1}x=si{n}^{-1}\sqrt{1-x^2}$

$\therefore sin\left(co{s}^{-1}x\right)=sin\left(si{n}^{-1}\sqrt{1-x^2}\right)=\sqrt{1-x^2}$

Similarly, $cos\left(si{n}^{-1}x\right)=cos\left(co{s}^{-1}\sqrt{1-x^2}\right)=\sqrt{1-x^2}$

$\therefore si{n}^{-1}\left(cos\right(si{n}^{-1}x\left)\right)+co{s}^{-1}\left(sin\right(co{s}^{-1}x\left)\right)$

$=si{n}^{-1}\left(\sqrt{1-x^2}\right)+co{s}^{-1}\left(\sqrt{1-x^2}\right)$

$=\frac{\pi }{2}$ ....$\left[si{n}^{-1}\right(\theta )+co{s}^{-1}(\theta )=\frac{\pi }{2}]$

Hence, the answer is option (A).