The Second Cosine Rule for Hyperbolic Triangles
For any htriangle ABC,
sin(B)sin(C)cosh(a) = cos(A) + cos(B) cos(C),
with similar formulae for cosh(b) and cosh(c).
Proof
We shall use some of the notation of the proof of the Sine Rule.
We put
α = cosh(a), β = cosh(b), γ = cosh(c), and
Δ^{2} = 1  α^{2}  β^{2}  γ^{2} + 2αβγ.
In that proof, we found that
Δ = sinh(b)sinh(c)sin(A)
= sinh(a)sinh(b)sin(C)
= sinh(a)sinh(c)sin(B)
From the Cosine Rule, we have
sinh(b)sinh(c)cos(A) = βγ  α,
sinh(a)sinh(c)cos(B) = αγ  β,
sinh(a)sinh(b)cos(C) = αβ  γ,
By appendix (1), sinh^{2}(x) = cosh^{2}(x)  1, so
sinh^{2}(a) = α^{2}  1,
Then we see that, after some cancelling,
(cos(A) + cos(B)cos(C))/sin(B)sin(C)
= ((αγ  β)(βα  γ) + (βγ  α)(α^{2}  1))/Δ^{2}.
After simplification, the top line turns out to be αΔ^{2},
and the result follows.

