Question

Evaluate
${\sin }^{-1}x+{\sin }^{-1}y={\cos }^{-1}${$\sqrt{(1-x^2)(1-y^2)}-xy$}

Answer 1

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

as ${\sin }^{-1}x{\cos }^{1}x=\frac{\pi }{2}$

$\therefore {\sin }^{-1}x=\frac{\pi }{2}-\cos x$

and ${\sin }^{-1}y=\frac{\pi }{2}-{\cos }^{-1}y$

$\therefore \frac{\pi }{2}-{\cos }^{-1}x+\frac{\pi }{2}-{\cos }^{-1}y$

$\Rightarrow \pi -\left({\cos }^{-1}x{\cos }^{-1}y\right)$

$\Rightarrow \pi -\left({\cos }^{-1}\right(xy-\sqrt{(1-x^2)(1-y^2)}\left)\right)$

$\Rightarrow {\cos }^{-1}\left\{\right(\sqrt{(1-x^2)(1-y^2)})-xy\}$

$RHS$

Hence proved