Plane trigonometry, by S.L. Loney.

CHAPTER XXIII.
EXPANSIONS OF sin n0 AND COS? 0. SERIES FOR sin 0
AND COS 0 IN POWERS OF 0.
274. BY the use of De Moivre's Theorem we can
obtain the expansion of cos nO and sin nO in terms of the
trigonometrical functions of 0.
For we have
cos nO + i sin n0 = (cos 0 + i sin O)n.
Since n is a positive integer, the Binomial Theorem
holds for (cos 0 + i sin 0)n.
Hence, by expanding, we have
cos nO + i sin nO = cosn 0 + n cos"-l 0. i sin 0
+ n(n- COSn-2 0. i2Sin2 0(+ ( 1)(n- os2) -30. sin 0...
+ 1. 1.2.3
Hence, since
i2=-1, i3=-i, i4=1, i5=i,...
we have
cos nO + i sin nO0 cosn 0 - n COsn-2 0 sin2 0
1.2
(n- 1) (n-2)(n-3) cos4 0 sin4 0 +...
1.2.3.4
+ i cosn-1 0 sin0 n (n 1) (- 2)con0 sin3 0 +..
~ [ s1.2.32
21-2