Difference of Even Powers of z + a and z - a

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

Then for all complex number $z$:
 * $\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \displaystyle \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$

Proof
From Factors of Difference of Two Even Powers:


 * $x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \displaystyle \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$

Substituting $z + a$ for $x$, $z - a$ for $y$, and $m$ for $n$ we get: