# 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, ..., an are complex numbers, no two of which differ by an integer multiple of π. Let

$A_{n,k} = \prod_{\begin{smallmatrix} 1 \le j \le n \\ j \neq k \end{smallmatrix}} \cot(a_k - a_j)$

(in particular, A1,1, 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:

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

## Notes and references

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