In the triangular pyramid shown we consider four triangles (two right triangles on a common hinge unit length normal to a striped triangle with a dihedral $\delta$ angle and the outer big yellow triangle containing compound angle $\gamma$).

We derive Cosine Rule in Spherical trigonometry indirectly avoiding representation of  sphere radius.

By applying Cosine Rule in striped triangle

$$ c^2= \tan^2\alpha+\tan^2\beta-2\tan\alpha \tan\beta \cos \delta $$

By applying Cosine Rule in larger yellow triangle containing compound angle $\gamma$

$$c^2= \sec^2\alpha+\sec^2\beta-2\sec\alpha \sec\beta \cos \gamma$$

Eliminate $c^2$ to simplify we get Cosine Rule in spherical trigonometry

$$\cos \gamma= \cos\alpha\cos\beta+ \sin \alpha \sin \beta  \cos \delta $$

We have used plane trig and embedded a pyramid into $\mathbb R ^3 :$

[![Triangles in SPH trig][1]][1]

Now how can we now draw the corresponding figure in hyperbolic geometry:

$$\cos \gamma= \cosh\alpha\cosh\beta+ \sinh \alpha \sinh \beta  \cos \delta \,? $$

Considering simpler cases visualization...  We can draw for right triangle $\delta= \pi/2$ the pyramid but how to at least draw it for hyperbolic geometry representation and result ?

$$ \cos \gamma= \cos\alpha\cos\beta \, \rightarrow  \cos \gamma= \cosh\alpha\cosh\beta \,? $$   

Thanks in advance for geometric considerations in hyperbolic geometry  without explicitly bringing in the pseudosphere.

Regards                     


  [1]: https://i.sstatic.net/fX2o1.png