The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

59] THE RIGHT1-ANGLED TRIANGLE 101
Thus sinh c = sinh a cosh 1......................... I.
(Hypothenuse, side, and opposite angle.)
From this formula, connecting the hypothenuse, a side, and
the opposite angle of any right-angled triangle, we can obtain
the relations between all the other elements, by using the
associated triangles of ~ 36.
We know that, starting with a right-angled triangle in
which the elements are
a, b, c, (X,,U),.........................(1)
we obtain successively triangles with the elements, a, 1, (R,2 l-,....................(2)
c', n', a, (X, 7- 3....................(3)
1,', -, ),...................(4)
'a, m, - ).....................(5)
From the second triangle
n', b, 1, ( 1 ->)'
we have sinh I = sinh m' cosh c
=i1 coshc, by ~ 58.
sinh m I
Therefore cosh c = sinh 1 sinh m.....................II.
(Hypothenuse and two angles.)
Also, from the same triangle (by I.),
sinh 1 = sinh b cosh a'
= sinh b coth a.
sinh b
Therefore tanl a= h.III)
Therefore tanh a= - -.............................II)
(Two sides and an angle.)
Now, since cosh c = sinh I sinh m,
sinh b sinh a
we have cosh c x
tanh a tanh b