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

Third Class. 43
46. Other Applications of Delambre's Formulae.
From the 3rd and 4th of Delambre's analogies, we have
/ c\ A + B c\ A-B
co s- cos -sn s- cos2 2 2
C ~ I C\; c ~ IO C\
cos 2 cos 90~ — sin2 cos 90 —
2 2/ 2 2/
From the first of these equations, we get
c\ c A + B C\
COS s-) +COS- COS, +Cos 90 —
___.___ 2_ _2 ___ _2
cos - - cos s- - cos 90- - - cos
or cot 2cot 2 = cot E tan (C - E); (177)
sin s- + sin cos +B cos(90 -
2, 2 2 2
sin(s-2) -sin2 cos 2 - cos (9O-2)
2 2
Hence, by division, and extracting the square root, we have
cot s = v/cot -E. tan { (4 - E) tan f (B - E) tan I ( C- E).
(179)
tan = v/tan E. tan (A - E)tan (B - ) cot }(C- E).
2
(180)
These simple and elegant proofs are due to Prouhet. See
Nouvelles Annales, 1856, p. 91.