The differential equation for which ${\sin }^{-1}x+{\sin }^{-1}y=C$ is given by

We have, ${\sin }^{-1}x+{\sin }^{-1}y=C$

This is one parameter family of curves. So, we differentiate it with respect to x to get 

$\frac{1}{\sqrt{1-x^2}}+\frac{1}{\sqrt{1-y^2}}\frac{dy}{dx}=0\Rightarrow \sqrt{1-y^2}dx+\sqrt{1-x^2}dy$=0