[732 
732] 
The numerator is 
A THEOREM IN SPHERICAL TRIGONOMETRY. 
.99 
■iy 
sin A 
sin a 
cos a cos 6 — cos c (cos 2 a + cos 2 6) + cos a cos b cos 2 c 
+ 1 — cos 2 c — (cos 2 a + cos 2 b) + cos a cos b. 2 cos c; 
viz. this is 
= cos a cos 6(1+ cos c) 2 - (cos 2 a + cos 2 b) (1 + cos c) + 1 - cos 2 c, 
having the factor 1 + cos c, which is also a factor of sin 2 c, = 1 — cos 2 c, in the 
denominator. We have, therefore, 
cos (A - B) = 
cos a cos 6(1+ cos c) — (cos 2 a + cos 2 6) + 1 — cos c 
(1 — cos c) sin a sin 6 
and the equation thus is 
(1 — cos c) (1 + cos a) (1 + cos 6) — {cos a cos 6 (1 + cos c) — (cos 2 a + cos 2 6) + 1 — cos c} 
= (1 + cos a) (cos a — cos 6 cos c) + (1 + cos 6) (cos 6 — cos c cos a), 
where each side is in fact 
= cos a + cos 2 a + cos 6 + cos 2 6 — cos c (cos a + cos 6) — 2 cos a cos 6 cos c ; 
and the second identity is thus proved. 
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ir values 
13—2