I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over \cos\left(\theta\right) - \cos\left(\alpha\right)}\,{\rm d}\theta\,, \qquad\qquad n\ \in\ {\mathbb Z}\,,\qquad 0\ \leq\ \alpha\ \leq\ 2\pi $$
There is singularity at $\alpha=\theta$ which increases its difficulty.
I tried complex analysis but couldn't solve it.please help me with method,also provide an answer with proof if you like, I would appreciate it.