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](https://doi.org/10.4064/aa118-3-4), arXiv:[math/0407306](https://arxiv.org/abs/math/0407306),

...but it isn't known if this is all instances of cosecant sums being zero.