1880.] Mr Glaisher, On theorems in elementary trigonometry. S23 



§ 6. It is worthy of remark that the theorem {A), which may 

 be briefly written 



n (sin a) + n (cos a) = 11 (sin a) + 11 (cos a), 



when squared reproduces itself. For, squaring each side, 



n (sin^ a) + n (cos" a) + ^ 11 (sin 2a) = a similar expression, 



that is 



IT (1 — cos 2a) + n (1 + cos 2a) + 211 (sin 2a) = a similar expression, 



viz. 



2 cos 2a cos 26 + 11 (cos 2a) +-11 (sin 2a) = a similar expression, 



which leads to (A) since, by § 4, 



X COS 2a cos 2b = X cos 2a cos 26'. 



§ 7. The other theorems referred to in the title are 

 sin a sin 6 sin c sin d = sin a' sin 6' sin c sin d' 



+ sin a" sin 6" sin c" sind" (B), 



cos a cos 6 cos c cos d = cos a' cos 6' cos c' cos (i' 



— sin a" sin 6" sin c" sin cZ" (C), 



sin^ a + sin^ 6 + sin^ c + sin^ d — sin^ a' — sin^ 6' — sin^ c — sin^ d' 



= — 4 sin a" sin 6" sin c" sin cZ" (D), 



cos a cos 6 cos c cos (i — sin a sin 6 sin c sin d 



= 1 — I (sin^a + sin^6' + sin^ c' + sin^ c?') {E); 



where a, 6', c, d' are as before, viz., 



a'=^{— a + b + c + d), 

 h'=^{ a-b + c + d), 

 c' = |( a+b-c + d), 

 cZ' = |( a+b + c — d); 

 and *" = i( a+6 + c + tZ), - 



6" = |( a + b — c — d), 

 c" = |( a — 6 + c — c?), 

 cZ"=i( a-6-c + (^). 



