If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$

Question

If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$

I am not finding any proper way even to express $y$ only in terms of $x$ too which could reduce bit complexity. Any help in proving this or validating will be highly appreciated.

Answer 1

HINT:

$$1=a\sin x\cot y+a\cos x\iff\cot y=\dfrac{1-a\cos x}{a\sin x}$$

Differentiate both sides wrt $x,$ $$-\csc^2y\cdot\dfrac{dy}{dx}=?$$

$$\csc^2y=1+\cot^2y=?$$