Hermite's cotangent identity explained

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 . Let

A _{n,k} = \prod _{\begin{smallmatrix} 1 \le j \le n \ j ≠ k \end{smallmatrix}} \cot(a _{k} - a _{j)}

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

\cot(z - a _{1) … \cot(z} - a _{n)} = \cos (n\pi)/(2) + \sum ^{n} _{k =1} 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