Sum over Integers of Cosine of n + alpha of theta over n + alpha

Theorem
Let $\alpha \in \R$ be a real number which is specifically not an integer.

For $0 < \theta < 2 \pi$:


 * $\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$

Proof
First we establish the following, as they will be needed later.

We have: