Hermite's cotangent identity

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 (in particular, A1,1, being an empty product, is 1).

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