Given a line $r$ and a (superior) semicircle perpendicular to $r$, and an arc $[AB]$ in the semicircle, I need to prove that
$$ \sinh(m(AB)) = \frac{\cos(\alpha)+\cos(\beta)}{\sin(\alpha)\sin(\beta)} \\ \cosh(m(AB)) = \frac{1 + \cos(\alpha)\cos(\beta)}{\sin(\alpha)\sin(\beta)} $$
where $\alpha$ is $\angle A'OA$ and $\beta$ is $\angle B'OB$.
The argument $m(AB)$ of $sinh(m(AB))$ and $cosh(m(AB))$ above is the hyperbolic Cayley-Klein hyperbolic metric of a hyperbolic segment
$$ m(AB) = \ln \left| \frac{AA' \cdot BB'}{BA' \cdot AB'} \right| $$
where $AB$ is the euclidean measure.
I tried writing the left side of both of them as the exponential definition. But it is so hard to manipulate the right side because always appear other segments like AO, BO and AB... Thanks.