Hermite's cotangent identity

In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite. Suppose a₁, ..., a_n are complex numbers, no two of which differ by an integer multiple of π. Let

$A_{n,k}=\prod _{\begin{array}{c}1\le j\le n\\ j\ne k\end{array}}\cot (a_k-a_j)$

(in particular, A_1,1, being an empty product, is 1). Then

$\cot (z-a_1)\cdots \cot (z-a_n)=\cos \frac{n\pi }{2}+\sum _{k=1}^n{A}_{n,k}\cot (z-a_k).$

The simplest non-trivial example is the case n = 2:

$\cot (z-a_1)\cot (z-a_2)=-1+\cot (a_1-a_2)\cot (z-a_1)+\cot (a_2-a_1)\cot (z-a_2).$