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

28 Connecting Sides and Angles of a Spherical Triangle.
EXERCISES.-V.
1. If D be any point of the side BC, prove that
cot AB sin DA C + cot A C sin DAB = cot AD sin BA C. (67)
2. cot ABC sin DC + cot A CB. sin BD = cot ADB. sin B. (68)
To prove 1, we have (Art. 29),
cot c sin AD = cot ) sin j + cos B. cos AD,
cot b sin AD = - cot D sin y + cos y cos AD. Eliminate cot D..
To prove 2, we have
cot AD sin m = cot B sin D + cos D cos m,
cot AD sin n = cot C sin D - cos D cos n. Eliminate cot AD...
3. If we describe a great circle B'D'C', with A as polar, equation (67)
gives us
tan DD'. sin B'C' = tan BB'. sin D'C' + tan CC'. sin B'D'. (69)
A
B/'D
Fig. 14.
a A
4. Prove 2cotma. sin 2 = (cot B + cot C) sin ( a. a). (70)
5. 2 os aot (. ) = ot cot C. (()
6.,, cot ma. tan = os (m. a). (72)
2 sin (B- C)