Question

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$ , then prove that ${\sin }^{-1}x={\cos }^{-1}y$

Answer 1

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