How to prove this quasi-geometric trigonometric series identity without induction

Question

$$\frac{2}{\sin{x}}\sum_{r=1}^{n-1} \sin{rx}\cos{[(n-r)y]} \equiv \frac{\cos{(nx)}-\cos{(ny)}}{\cos{x}-\cos{y}} - \frac{\sin{(nx)}}{\sin{x}}$$

The identity can be tediously proven using the Axiom of Induction.

I am looking for other means of proving this identity.

Answer 1

Hint: First use $2\sin(x)\cos(y)=\sin(x+y)+\sin(x-y)$, then apply $2i\sin(u)=e^{iu}-e^{-iu}$.