324 Mr Glaisher, On theorems in elementary trigonometry. [Feb. 9, 



The five theorems may be written more compendiously as 

 follows : 



n (sin a) + n (cos a) =Tl (sin a) + 11 (cos a) {A), 



n(sina) = n(sinfO+n(sina") {B), 



n (cos a) = n (cosa) — n (sin a") ( C), 



2 sin^ a — 2 sin^a = — 411 (sin a") {B), 



n (cos a) — n (sin a) = 1 — 1 2 (sin^ a) {E). 



The first theorem, {A), is derivable by addition from [B) 

 and (C). 



§ 8. These theorems may be proved directly without difficulty : 

 for if a — a + ^, 



h = a-/3, 



c = 7 + S, 



d = j-S, 



then a = y — /3, a" = a + y, 



&' = 7 + yS, b" = a — <y, 



c = a— S, c" — (3 + 8, 



d' = 0. + 8, d" = — S, 



and the theorems become 



sin (a + 13) sin (a. — /3) sin (7 + S) sin (7 — S) 



+ cos (a + /3) cos (a — 13) cos (7 + 8) cos (7 — 8) 

 = sin (7 + /9) sin (7 — j3) sin (a + 8) sin (a — 8) 



+ cos (7 + /3) cos (7 - /3) cos (a + 8) cos {a - 8) ... (^'), 

 sin (a + /3) sin (a — /3) sin (7 + 8) sin (7 — 8) 



= sin (7 + ^) sin (7 - /3) sin [a + 8) sin (a - S) 

 + sin (a + 7) sin {u - 7) sin (/3 + g) sin {/3 - 8) ... (5'), 

 cos (a + yS) cos (a — yS) cos (7 + S) cos (7 - 8) 



= cos (7 + /3) cos (7 - j3) cos (a + 8) cos (a - 8) 



-sin(a+7) sin(a-7) sin (/3 + 3) sin (/3 - S) (C), 



sin' (a + yS) + sin' (a - /S) + sin' (7 + S) + sin' (7 - 8) 



- sin- (7 + yS) - sin' (7 - /3) - sin' (a + S) - sin' (a - 8) 

 = - 4 sin (a + 7) sin (a - 7) sin (/3 + S) sin (/3 - S) (D'), 



