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

26 Connecting Sides and Angles of a Spherical Triangle.
EXERCISES.-IV.
1. If a, b, c be the sides of a spherical triangle, a', b', c' the sides of its
supplemental triangle, prove
sin a: n: sin c::sin a': sin b': sin c'. (48)
8n3
2. Prove that sinA sinB sin C = s2. si.bin (49)
sin2a sm2 b sm2c
3. Prove that
tan (A + B): tan (A- B):: tan (a+ b): tan (a-b). (50)
4. Prove that
tan (a + a): tan ~ ( - a): tan (B + b): tan (B- b). (51)
5. If the bisector AD of the angle A of a spherical triangle divide the
side BC into the segments CD = b', BD = c', prove
sin b: sin c:: sin b': sin '. (52)
6. If the bisector of the exterior angle, formed by producing BA through
A, meet the base BC in D', and if BD' = c", CD' = b", prove that
sin b: sin:: sin b": sin c". (53)
7. Prove that
cot A: cot ~ B: cot:: sin (s-a): sin ( - b): sin (s - c). (54)
8. If D be any point in the side BC of a spherical triangle,
sin BD sin BAD sin C
(55)
sin CD = sin CAD sin B'
Cor.-If D be the middle point,
sin BAD sin B sinb
_ - ~~= ~ ~ ~(56)
sin CAD sin C sin ce
9. Prove that sin a sin ha = sin b sin hb = sin c sin hc = 2n. (57)
10. Given the base of a spherical triangle and the norm of the sides,
prove that the locus of the vertex is a small circle.
11. If mb = m, prove that either b = c, or
sin2 a = cos2 b + cos2 ~ c + cos ~ b cos c.
12. If a great circle touch two small circles, whose spherical radii are
p, p', and distance between their poles = 8, and if r denote the arc between
the points of contact, prove
Sin2 rsin2 S - sin2 (p - p')
coS p cos p'