Question

Solve for $x:{\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x$.

Answer 1

The given equation is
${\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x$
$\Rightarrow {\sin }^{-1}x+{\sin }^{-1}(1-x)=\frac{\pi }{2}-{\sin }^{-1}x$
$\Rightarrow {\sin }^{-1}(1-x)=\frac{\pi }{2}-2{\sin }^{-1}x$ ...... (i)
Let ${\sin }^{-1}x=y$
$\Rightarrow x=\sin y$

Therefore, from (i), we get
${\sin }^{-1}(1-x)=\frac{\pi }{2}-2y$
$\Rightarrow 1-x=\sin (\frac{\pi }{2}-2y)$
$\Rightarrow 1-x=\cos 2y$
$\Rightarrow 1-x=1-2{\sin }^{2}y$
$\Rightarrow 1-x=1-2x^2$
$\Rightarrow 2x^2-x=0$
$\Rightarrow x(2x-1)=0$
$\Rightarrow x=0,\frac{1}{2}$

Since, both these values satisfy the given equation.

Hence, the solutions of the given equation are $x=0,\frac{1}{2}$.