I've stumbled across an identity that I think holds, but can't prove it. The identity is:
$$ \sum_{i=1}^{N} \cot(\phi_i - \phi_{i+1}) \cot(\phi_i - \phi_{i-1}) = -1 $$
The angles $\phi_i$ are some arbitrary set of angles, with the only constraint being that $\phi_i - \phi_{i+1}$ does not equal $0$ or $\pi$. Addition of the indices is modular, so that $\phi_{N+1} = \phi_{1}$.
The formula comes from some normalisation conditions in quantum mechanics, and it arose while checking a result that I've derived otherwise, which is my reason for believing it stands. It can be shown straightforwardly using Hermite's cotangent identity if $N=3$, but I'm unable to generalise further.
Any ideas? Thanks for your help.
https://en.wikipedia.org/wiki/Hermite%27s_cotangent_identity
Does this Wikipedia article answer your question?