Cosine Law Duality in Hyperbolic Trigonometry From setting up a hyperbolic triangle with hyperbolic side length $a,b,c$ and corresponding angles $A,B,C$, it is not hard to prove the following law of cosine: 
$$\cos A= \frac{\cosh b \cosh c -\cosh a}{\sinh a \sinh b}$$
Since we have $3$ formula for angles $A,B,C$, with brute force, it is possible to show the following identity:
$$\cosh a =\frac{\cos B \cos C + \cos A}{\sin B \sin C}$$
Comparing the formula above, it seems like there is a duality between hyperbolic distance and angles. Are there any deeper geometric implication in phenomenon?

I explain this phenomen with “complex hexagonometry”.
Let $z_{1},z_{2},z_{3},w_{1},w_{2},w_{3}$ be complex numbers. Let (i,j,k) be the cyclic permutation of (1,2,3). Complex hexagonometry is the involutory transformation of 3 functions of 3 variables defined as follows:
Cosine formulas:
    $cosh(w_{i})=(cosh(z_{i})+cosh(z_{j})cosh(z_{k}))/(sinh(z_{j})sinh(z_{k}))$. 
 Its inverse cosine formulas:
    $cosh(z_{i})=(cosh(w_{i})+cosh(w_{j})cosh(w_{k}))/(sinh(w_{j})sinh(w_{k}))$.
Sine formulas:
    $sinh(w_{1})/sinh(z_{2})=sinh(w_{2})/sinh(z_{2})=sinh(w_{3})/sinh(z_{3})$. 
We can deduce above 2nd and 3rd formulas from 1st formulas.
We can specialize complex hexagonometry to hyperbolic trigonometry.
We set $z_{1}=a_{1}+\pi i,z_{2}=a_{2}+\pi i,z_{3}=a_{3}+\pi i,w_{1}=-A_{1}i,w_{2}=-A_{2}i,w_{3}=-A_{3}i$,where $a _{1},a_{2},a_{3} $ are positive real numbers and $A_{1},A_{2},A_{3}$ are inner angles.
We can specialize complex hexagonometry to spherical trigonometry.
We set $z_{1}=(a_{1}+\pi)i,z_{2}=(a_{2}+\pi)i,z_{3}=(a_{3}+\pi)i,w_{1}=-A_{1}i,w_{2}=-A_{2}i,w_{3}=-A_{3}i$  where$ a_{1},a_{2},a_{3}$ are positive real numbers and are $A_{1},A_{2},A_{3}$ are inner angles.
Complex hexagonometry does not discriminate distances and angles.
