98 
A THEOREM IN SPHERICAL TRIGONOMETRY. 
[732 
that is, 
mn O 
— sin (A — B) = j( cos a ~ cos &) 
cos c 
sin C sin c 
sin cl — cos c 
(cos a — cos ò) ; 
or, what is the same thing, 
sin (J 
— tan -|c sin (A — B) = —(cos a — cos b). 
Here cosa —cosò is = (1 + cos a) — (1 + cos b) ; substituting for S * U ^ successively 
sin c 
and the right-hand side is 
sm 6 
1 + cos a . . 1 4- cos b . „ 
= . sm A ; sm B, 
sm a sm b 
= cot ^a sin A — cot b sin B; 
whence, multiplying each side by tan^atan-|6, we have the relation in question. 
For the second identity which is 
tan {1 — tan\ct tan b cos {A — 5)} = tan \ b cos A + tan ^a cos B ; 
if on the right-hand side we substitute for cosH, cosi? their values 
cos a — cos b cos c . cos b — cos a cos c 
7 7 7 and 7 7 , 
sm b sm c sm a sm c 
the right-hand side becomes 
sm a 
cos a — cos b cos c cos b — cos a cos c 
+ 
1 
sin c I 1 + cos b ' 1 + cos a 
whence, multiplying the whole equation by sinc(1 + cosa) (1 + cosò), it becomes 
(1 — cos c) {(1 + cos a) (1 + cos b) — sin a sin ò cos {A — J5)} 
= (1 + cos a) (cos a — cos ò cos c) + (1 + cos b) (cos ò — cos c cos a). 
We have here 
, . i • -r> (cos a — cos b cos c) (cos ò — cos c cos a) + □ 
cos (A — B) = cos A cos B + sm A sm B = . ■■■-.' =—t , 
v sm- 2 c sm a sm ò 
by substituting for cosH, cos B their foregoing values, and for sin A, sin B their values 
vn vo 
sin ò sin c ’ sin a sin c 
, where 
□ = 1 — cos 2 a — cos 2 b — cos 2 c + 2 cos a cos ò cos c.