Integral over 2 pi of Cosine of m x by Cosine of n x

Theorem
Let $m, n \in \Z$ be integers.

Let $\alpha \in \R$ be a real number.

Then:
 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \begin {cases}

0 & : m \ne n \\ \pi & : m = n \end {cases}$

That is:
 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \pi \delta_{m n}$

where $\delta_{m n}$ is the Kronecker delta.

Proof
Let $m \ne n$.

When $m = n$ we have: