Plane trigonometry, by S.L. Loney.

350 TRIGONOMETRY.
Picking out the required coefficients as in Art. 293, we
have
n+1 n-1
sin...nO. 2 21' 2c
2- '- j(- cos )
sin n 1- 2 2 COS 0)2
n+3 n+1 n-1 n-3
n +3 * - - * 3
+ n3 2 2 2 -2 (-2cos4+...
1.2.3.4
+ (2 cos O)-1
Hence, finally, when n is odd, we have,-1 sin nO n2- 12 0 (_2-2)(2-32),
(n2 - 12) (n2 - 32) ( l2 - 52) cos
_ (.~ r)(~l~ -~ 3~)( os, o -......
16'
n-l
+ (-1) 2 (2 cos O)n-1.........(2).
Case II. Let n be even, so that n - 1 is odd.
The lowest term in (1) which gives any coefficient of
xn-l is then that for which
n
r= -
2'
Hence, in this case,
sin= coefficient of x-1 in 1 - x (x - 2 cos 0) +...
sin 0
n n n nL + n
+ (-1)2 2( - 2 cos0)2+ (-1)2 m2 (x - 2 cos 0)2
( +2 -+2 (+2
++ (- 1)2 x+2 (x - 2 cos 0)2) +.
+ (- 1)n-1 'n-i (x - 2 COS 09)n-1 +.......