Question

Solve:
${\sin }^{-1}x+{\sin }^{-1}\sqrt{1-x^2}$

Answer 1

REF.Image.

$I=si{n}^{-1}x+si{n}^{-1}\sqrt{1-x^2}$

In $△ABC$

$\Rightarrow sinA=\left(\frac{x}{1}\right)$

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

$sinC=\frac{\sqrt{1-x^2}}{1}$

$\Rightarrow C=si{n}^{-1}\left(\sqrt{1-x^2}\right)$

$\therefore A+C=si{n}^{-1}\left(x\right)+si{n}^{-1}\left(\sqrt{1-x^2}\right)$

$A+B+C={180}^{\circ }\Rightarrow$

$A+90+C={180}^{\circ }\Rightarrow A+C={90}^{\circ }=\pi /2$

$\therefore si{n}^{-1}\left(x\right)+si{n}^{-1}\left(\sqrt{1-x^2}\right)=\pi /2$