Question

Prove that ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$ for $|x|\le 1$.

Answer 1

Let $A={\sin }^{-1}x$......(1) and $B={\cos }^{-1}x$.....(2).

From (1) and (2) we get,

$x=\sin A=\cos B$

or, $\sin A=\sin ({90}^o-B)$

or, $A=\frac{\pi }{2}-B$

or, $A+B=\frac{\pi }{2}$

or, ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$.