Plane trigonometry, by S.L. Loney.

CHAPTER XXIV.
EXPANSIONS OF SINES AND COSINES OF MULTIPLE ANGLES,
AND OF POWERS OF SINES AND COSINES.
[On a first reading of the subject the student is recommended
to omit from the beginning of Art. 293 to the end of the chapter.]
287. IN this chapter we shall shew how to expand
powers of cosines and sines of an angle in terms of cosines
and sines of multiples of that angle, and also how to
express cosines and sines of multiple angles in terms of
powers of cosines and sines.
288. Let x = cos 0 + i sin 0, so that
1 1 cos - i sin 0
= cos 0 - i sin 0.
x cos 0 + isin 0 cos2 + sin2 0
1
Hence x + - = 2 cos 0,
1
and - -= 2i sin 0.
Also, by De Moivre's Theorem, we have
xn = cos nO + i sin nO,