There are other identities that are relevant, but are less systematically understood. For example,
$$\Gamma \left(\frac{1}{7}\right) \Gamma \left(\frac{6}{7}\right)=\Gamma \left(\frac{3}{7}\right) \Gamma \left(\frac{4}{7}\right)+\Gamma \left(\frac{2}{7}\right) \Gamma \left(\frac{5}{7}\right).$$
There's a known generalization of this with 7 replaced by $2^k-1$, see my paper with Ron Graham:
- Ron Graham, Kevin O'Bryant, A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture, Acta Arith. 118 (2005), no. 3, 283–304, doi:10.4064/aa118-3-4, arXiv:math/0407306,
...but it isn't known if this is all instances of cosecant sums being zero.
