Plane trigonometry, by S.L. Loney.

346 TRIGONOMETRY.
EXAMPLES. LI.
Prove that
1. si5-sin 0 = 10 [sin -5 sin 310 sin 0].
2. cos9 0 = 26 [cos 90 + 9 cos 70 + 36 cos 50 +~84 cos 30 + 126 cos 0].
3. cos10 0=
2 [cos 100 + 10 cos 80 + 45 cos 60 + 120 cos 40 + 210 cos 20 + 126].
512
4. sin8= 1 [cos 80 -8 cos 60 +28 cos 40 -56 cos 20 + 35].
5. sin9 0 = 26 [sin 90 - 9 sin 70 + 36 sin 50 - 84 sin 30 +126 sin 0].
sin nO.
*293. To express sin in a series of descending
powers of cos 0.
If x be < 1, we have
sin 0
12s +x22l=- sin 0 + x sin 20 + 'x sin 30 +..
1 - 2x cos 0 + x2
- + x -' sin +... ad inf..........(1).
This may be shewn by multiplying each side by
1 - 2x cos 8 + x2,
when it will be found that the right-hand member will
reduce to sin 0.
Another proof will be found in Art. 358.
Equating coefficients of xn-l in (1), we have
sin nO
sn n = coefficient of xn-1 in [1 - 2x cos 0 + X2]-1
sinO0
= coefficient of xn-1 in [1- x (2 cos 0 - x)] —
coefficient of xn-l in
1 + x (2cos - x)+ x (2cos - x)2 +....
+- x-3 (2 cos 0 - x)n + x n-2 (2 cos 0 - x)n —
+x&1-1 (2 cos 0 - x)_-1 + xn (2 cos 0 - x)n +...... (2).