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.