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:

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

For simpler cases the first relation can be drawn for right trianglee $\delta= \pi/2$..

but how to at least draw the latter pyramid yielding hyperbolic geometry 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