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 $\gamma$).

This is intended to derive Cosine Rule in Spherical trigonometry indirectly avoiding representation of sphere radius.

How to draw this figure in hyperbolic geometry,in order to arrive at

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

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

Now how can one 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