if $x\sin (a+y)=\sin y\text{then}\frac{dy}{dx}$ is equal to

$x=\frac{\sin y}{\sin (a+y)},$then differentiate

$\Rightarrow \frac{dx}{dy}=\frac{\sin (a+y)\cos y-\sin y \cos (a+y)}{{\sin }^2(a+y)}$

$\Rightarrow \frac{dx}{dy}=\frac{\sin (a+y-y)}{{\sin }^2(a+y)}$

$\Rightarrow \frac{dy}{dx}=\frac{{\sin }^2(a+y)}{\sin a}$