A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

First Class. 37
17. Prove that sin2 = sin2 cos cos os22 sin2-. (130)
2 2 2 2 2
A B
18.,, sin(a - b) = sin atan - sin b tan. (131)
2 2
19.,, sin(c- b) = sin (b + c) tan2 -, (132)
20.,, sin (c - a) = cos a sin b tan B. (133)
21. Prove that in an equilateral spherical triangle
2sin A = sec 1 a. (134)
22. If the opposite angles A, C of a spherical quadrilateral ABCD be
right, and if the sides AD, BC produced meet in E, prove
tanAE. tanDE = tan BE. tan CE. (135)
23. If the internal and external bisectors of the angle A of a spherical
triangle meet the base in D, D', prove
t sin2b - sin2 a
cot DD' = Sm (136)
2 sin a sin b sin c
24. Given the base of a spherical triangle, and the sum of the base angles,
prove that the external bisector of the vertical angle passes through a given
point.
25. If CC' be the median from the right angle of a right-angled triangle,
prove that
sin2ina in2b = (2 cos - sin CC')' (137)
26-34. If p be the perpendicular from the right angle C on the hypotenuse, proveC
B
Fig. 17.
1~. cot2p = cot2a + cot2b. 2~. cos2p = cos2A + cos2B. (138)
3~. tan2a = + tan a' tan c, tan2 b = + tan b' tan c. (139)
4~. tan2a: tan2b:: tana': tanb'. (140)