Hermite's cotangent identity
# Hermite's cotangent identity

In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite. Suppose $a_1,$ ..., $a_n$ are complex numbers, no two of which differ by an integer multiple of π. Let

    $A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})$

(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:

    $\cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).\,$
