Compute $\int_0^\pi dx [(\sin(nx)-\sin((n-1)x))/\sin(x/2)]^2$

Question

Is is possible to evaluate analytically the integral

$\int_0^\pi dx \left[\frac{\sin(nx)-\sin((n-1)x)}{\sin(x/2)}\right]^2$

where $n\in\mathbb N$?

Thanks in advance.

Answer 1

Hint: Use the formula $$\sin a-\sin b=2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$$