Hermite's cotangent identity

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 Template:Pi. Let

A_{n, k} = \prod_{\begin{matrix} 1 \leq j \leq n \\ j \neq k \end{matrix}} \cot (a_{k} - a_{j})

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

\cot (z - a_{1}) \dots \cot (z - a_{n}) = \cos \frac{n π}{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

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