The Tangent Formula for Hyperbolic Triangles
If the htriangle ABC has a right angle at A, then
tan(B) = tanh(b)/sinh(c), and
tan(C) = tanh(c)/sinh(b).
Proof
By Pythagoras's Theorem, cosh(a) = cosh(b)cosh(c),
so that cosh(b) = cosh(a)/cosh(c).
By the Sine and Formulae,
sin(B) = sinh(b)/sinh(a), and
cos(B) = tanh(c)/tanh(a).
Thus
tan(B)

= sinh(b)tanh(a)/sinh(a)tanh(c)



= sinh(b)cosh(c)/cosh(a)sinh(c)

as tanh(x) = sinh(x)/cosh(x) 

= sinh(b)/cosh(b)sinh(c) 
as cosh(a) = cosh(b)cosh(c) 

= tanh(b)/sinh(c) 
as tanh(x) = sinh(x)/cosh(x) 
The proof for tan(C) is similar.
