Generalized half-angle formulas - Hyperbolic version

In Mathoverflow I wonder about the possibility of finding non-Euclidean versions of the generalized half-angle formulas $[1]$. The Russian mathematician Alexander Mednykh has kindly answered my question. Specifically, he has derived the hyperbolic version. Look at the following link:

Note: Generalized half-angle formula - Hyperbolic version

Reference
$[1]$ E. A. José García, Two Identities and their Consequences, MATINF, 6 (2020) 5-11.