  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
H\"older continuity result for quasiregular mappings with respect to the visual
angle metric.
