Question

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

Answer 1

Let,

${\sin }^{-1}x=A,{\sin }^{-1}(1-x)=B,{\cos }^{-1}x=C$

$⟹\sin A=x,\sin B=(1-x),\cos C=x$

$\cos A=\sqrt{1-x^2},\cos B=\sqrt{1-(1-x{)}^2}=\sqrt{2x-x^2},\sin C=\sqrt{1-x^2}$

Given, $A+B=C$

$\sin (A+B)=\sin C$

$\sin A\cos B+\cos A\sin B=\sin C$

$x\left(\sqrt{2x-x^2}\right)+\left(\sqrt{1-x^2}\right)(1-x)=\sqrt{1-x^2}$

$x\left(\sqrt{2x-x^2}\right)=x\left(\sqrt{1-x^2}\right)$

This has two solutions,

(i) $x=0$

(ii) $\sqrt{2x-x^2}=\sqrt{1-x^2}$

$2x-x^2=1-x^2$

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