Question

Find the value of x which satisfy the equation ${sin}^{-1}x+{sin}^{-1}(1-x)={cos}^{-1}x$

Answer 1

From the given equation, we have $\sin ({sin}^{-1}x+{sin}^{-1}(1-x\left)\right)=\sin \left({cos}^{-1}x\right)$

$\Rightarrow \sin \left({\sin }^{-1}x\right)\cos \left({\sin }^{-1}\right(1-x\left)\right)+\cos (-si{n}^{-1}x)\sin \left({\sin }^{-1}\right(1-x\left)\right)=\sin \left({\cos }^{-1}x\right)$

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

$\Rightarrow x\sqrt{2x-x^2}+\sqrt{1-x^2}(1-x-1)=0$

$\Rightarrow x(\sqrt{2x-x^2}-\sqrt{1-x^2})=0$

$\Rightarrow x=0$ or $2x-x^2=1-x^2$

$\Rightarrow x=0$ or $x=\frac{1}{2}$