Cotangent Additive Formula

Theorem

Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.

Let $z \in \C$ such that $\map \sin {n z} \ne 0$

Then:

$\ds \map \cot {n z} = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \cot {z + \frac {k \pi} n}$

where $\cot$ denotes the cotangent function.

$\ds \map \sin {n z}$

$=$

$\ds 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$

$\ds \leadsto \ \ $

$\ds \map \ln {\map \sin {n z} }$

$=$

$\ds \paren {n - 1} \ln 2 + \sum_{k \mathop = 0}^{n - 1} \map \ln {\map \sin {z + \frac {k \pi} n} }$

$\ds \leadsto \ \ $

$\ds n \map \cot {n z}$

$=$

$\ds \sum_{k \mathop = 0}^{n - 1} \map \cot {z + \frac {k \pi} n}$

$\ds \leadsto \ \ $

$\ds \map \cot {n z}$

$=$

$\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \cot {z + \frac {k \pi} n}$

dividing by $n$

$\blacksquare$