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

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.