Visual angle metric in the upper half plane

We prove an identity which connects the visual angle metric $v_{{\mathbb{H}}^2}$ and the hyperbolic metric ${\rho }_{{\mathbb{H}}^2}$ of the upper half plane ${\mathbb{H}}^2$. The proof is based on geometric arguments and uses computer algebra methods for formula manipulation. We also prove a sharp Hölder continuity result for quasiregular mappings with respect to the visual angle metric.