Whether the hyperbolic trigonometry rule holds for all models

I am reading a paper concerning hyperbolic geometry. It represents some results like the hyperbolic cosine rule. Consider a hyperbolic triangle with side lengths $a$, $b$, $c$ and angles $\alpha$, $\beta$, $\gamma$. Then the hyperbolic cosine rule says $\cosh{a} = \cosh{b} \cosh{c} - \sinh{b} \sinh{c}\cos{\alpha}$.

My question is, does this rule works for all models of hyperbolic plane? Including the Poincare's model for simply connected domain and Klein's model. And also, what is the relationship between different models of the hyperbolic plane.