Question

Solve the equation for x:
${\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x,x\ne 0$

Answer 1

Let $x=\sin y=\cos (\frac{\pi }{2}-y)$

$1-x=1-\sin y$

According to the question ,

${\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x$

$y+{\sin }^{-1}(1-\sin y)=\frac{\pi }{2}-y$

$\sin (\frac{\pi }{2}-2y)=1-\sin y$

$\cos 2y=1-\sin y$

$1-2{\sin }^{2}y=1-\sin y$

$\sin y(1-2\sin y)=0$

$\sin y=0$

$x=0$

and

$\sin y=\frac{1}{2}$

$x=\frac{1}{2}$