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.
Yes. This rule has nothing to do with the model being used -- it has only to do with the geometry of the hyperbolic plane itself.
In general, different models of the hyperbolic plane are isometric, meaning they are just different descriptions of the same Riemannian manifold. That is, different models of the hyperbolic plane are very similar to different map projections of the spherical earth -- none of them is quite "right" for both distances and angles, since you can't actually embed a sphere on a plane, but you can use any of them to do spherical geometry if you're willing to keep in mind what concepts like "distance", "angle", and "geodesic" actually correspond to in the model.