Hermite's cotangent identity

Not to be confused with Hermite's identity, a statement about fractional parts of integer multiples of real numbers.

In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite.^[1] Suppose a₁, ..., a_n are complex numbers, no two of which differ by an integer multiple of π. Let

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

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

${\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}$

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

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

Notes and references

1. Warren P. Johnson, "Trigonometric Identities à la Hermite", American Mathematical Monthly, volume 117, number 4, April 2010, pages 311–327