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

Let $m \in \Z$ be an integer such that $m > 1$.
Then for all complex number $z$:
:$\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \ds \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$

Proof:
From Factors of Difference of Two Even Powers:
:$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \ds \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$
Substituting $z + a$ for $x$, $z - a$ for $y$, and $m$ for $n$ we get:
{{begin-eqn}}
{{eqn | o = 
      | r = \paren {z + a}^{2 m} - \paren {z - a}^{2 m}
      | c = 
}}
{{eqn | r = \paren {\paren {z + a} - \paren {z - a} } \paren {\paren {z + a} + \paren {z - a} } \prod_{k \mathop = 1}^{m - 1} \paren {\paren {z + a}^2 - 2 \paren {z + a} \paren {z - a} \cos \frac {k \pi} m + \paren {z - a}^2}
      | c = 
}}
{{eqn | r = \paren {2 a} \paren {2 z} \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + 2 a z + a^2 - 2 \paren {z^2 - a^2} \cos \frac {k \pi} m + \paren {z^2 - 2 a z + a^2} }
      | c = 
}}
{{eqn | r = 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 + 2 a^2 - 2 \paren {z^2 - a^2} \cos \frac {k \pi} m}
      | c = 
}}
{{eqn | r = 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 \paren {1 - \cos \frac {k \pi} m} + 2 a^2 \paren {1 + \cos \frac {k \pi} m} }
      | c = 
}}
{{eqn | r = 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 \paren {2 \sin^2 \frac {k \pi} {2 m} } + 2 a^2 \paren {2 \cos^2 \frac {k \pi} {2 m} } }
      | c = {{Corollary|Double Angle Formula for Cosine|1}} and {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | r = 4 a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} 4 \sin^2 \frac {k \pi} {2 m} }
      | c = {{Defof|Cotangent|subdef = Real Function}}
}}
{{eqn | r = 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = 1}^{m - 1} \map \sin {\pi - \frac {k \pi} {2 m} } }
      | c = Sine of Supplementary Angle
}}
{{eqn | r = 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {\paren {2 m - k} \pi} {2 m} } \sin \frac {m \pi} {2 m}
      | c = Sine of Right Angle
}}
{{eqn | r = 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^m \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = m + 1}^{2 m - 1} \sin \frac {k \pi} {2 m} }
      | c = Translation of Index Variable of Product
}}
{{eqn | r = 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{2 m - 1} \sin \frac {k \pi} {2 m} }
}}
{{eqn | r = 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\frac {2 m} {2^{2 m - 1} } }
      | c = Product of Sines of Fractions of Pi
}}
{{eqn | r = 4 m a z \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} }
}}
{{end-eqn}}
{{qed}}