Question

Prove that : ${\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$

Answer 1

Suppose that
${\sin }^{-1}x=\alpha \Rightarrow x=\sin \alpha$
and ${\sin }^{-1}y=\beta \Rightarrow y=\sin \beta$
Now $\sin (\alpha +\beta )=\sin \alpha .\cos \beta +\cos \alpha .\sin \beta$
$=\sin (\alpha +\beta )=\sin \alpha .\sqrt{1-{\sin }^2\beta }+\sqrt{1-{\sin }^2\alpha }.\sin \beta$
$=\sin (\alpha +\beta )=x.\sqrt{1-y^2}+y.\sqrt{1-x^2}$
$=\alpha +\beta ={\sin }^{-1}[x.\sqrt{1-y^2}+y.\sqrt{1-x^2}]$
$={\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$.