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

87, 881 TRIGONOMETRICAL FORMULAE

149

NOTE. At several points in this argument we have assumed
that the segments concerned are less than,.
Once the fundamental theorem has been proved for triangles
in which this condition is satisfied, it can be extended by
analysis to all other cases.
~ 88. The remaining formulae are easily obtained:
b C
To prove tan = cos A tan..................... (2)
k k.
Let ABC be any right-angled triangle, with C a right angle.
E/^ /
Z \/,
A r D C
FIG. 105.
Take any point D on AC, and join BD.
Draw DE perpendicular to AB.
Let AE=p, ED=q, AD=r, and BD=l.
Then, from the triangle ABC, we have
I a b - \
cos C =S Cos cos
(a b ' a. b. r
= Co cos cos cos + cos s
c r a. b.
= cos - co -+ os - sin sin.
k *1c c k k1
Also, from the triangle BDE, we have in the same way
I c q. p. c
1 c r q p csin sin -
cos - = CS - co s + cos - sin
aTherefor e cos si si r q..
Therefore cos sisin - sin = co - sin sini k k ik ki