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

62] A FUNDAMENTAL FORMULA 10

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With the notation of this section, we have
cosf(m + l) = tanh A (, + Al)
cosh c cosh cl - cosh (b + bl)
sinh c sinh c
cosh b cosh b, sinh b sinh b.
= coth c coth G 1 - -
1sih c sinh c s inh c sinh c s
But we know that
tanh a = tanh c tanh mn, [~ 59, VI.]
i.e. tanh a = tanh c cosf(u).
Similarly tanh a = tanh c cos/f(/,).
Therefore coth c coth c = coth2a cosf/(,) cosf(,ul).
Further, from ~ 59, I., we obtain
sinh b 1
sinh c cosh m= sinf(),
sinh b_ 1
sinh cosh c m
Therefore sinh c sinh b1 = sin/f() sinf(l1).
sinl c sinh c1
We are left with the term
cosh b cosh b.
sinh c sinh c1
But, from ~ 59, VI. and IV., we have
tanh m cosh c cosh b
sinh a sinh c cosh a sinh c
Therefore cosh b cosh b = cosf (u) cosf(m,)
sinh cinc sinh sinh a
Thus we obtain
cosf(/ +,1) = coth2a cosf(m) cosf(/()
- cosech2a cos f()'cosf(/z,) - sinf(/,) sinf((l)
= cosf(/u) cosf(/xl) - sinf(/A) sinf(/l)
= cos [/(Iu) +f/(l)].