Product of Cotangents of Fractions of Pi

Theorem
Let $m \in \Z$ such that $m > 1$.

Then:
 * $\displaystyle \prod_{k \mathop = 1}^{m - 1} \cot \frac {k \pi} {2 m} = 1$

Proof
We have:

That means:
 * $(1): \quad \dfrac \pi 2 - \dfrac {k \pi} {2 m} = \dfrac {\paren {m - k} \pi} {2 m}$

Let $m$ be odd.

Then $m - 1$ is even, and so:

Now suppose $m$ is even.

Then $m - 1$ is odd, and so: