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

First Class. 27
13. If AD be the median, prove
cot DB =cos a (cosb - cosc) (59)
cot2ADB~ 2n ()
14. Prove that
A A A
cot (ina, a) sec a + cot (mb, b) sec b + cot (m, c) sec c = 0. (60)
Case III.-Two Sides and two Angles, one of which is
contained by the Sides.
30. If we multiply equation (15) by cos b, and substitute the
result in (13), we get
cos a = cos a cos2 b + sin a sin b cos b cos C + sin b sin c cos A.
Hence, transposing cos a cos2 b, substituting sin2 b for 1 - cos2 b,
and dividing by sina sinb, we get
cot a sin b = cos b cos C +. cos;
sin a
sin C sin c
and substituting, for
sinA sin a'
we get cot a sin b = cotA4 sin C + cos 0 cos b. (61)
This important formula may be enunciated as follows, calling
a the first side, and b the second:-The cot of the first, by the
sine of the second, is equal to the cot of the angle opposite to the
first, by the sine of the contained angle, plus the cos of the contained
angle, by the cos of the second side.
By interchanging letters in (61), we get
cot a sin c = cot A sin B + cos B cos c. (62)
cot b sin a = cot B sin C + cos C cos a. (63)
cot b sin c = cot B sin A + cos A cos c. (64)
cot c sin a = cot C sin B + cos B cos a. (65)
cot c sin b = cot C sin 2 + cos A cos b. (66)