Plane trigonometry, by S.L. Loney.

EXPANSION OF COSn 0.

34.

and - = cos nO - i sin nO,
so that x7 + = 2 cos no,
Xn
n
and n- -- 2isin n.
xn
289. To expand cos 0 in a series of cosines of multiples
of 0, n being a positive integer.
From the previous article we have
(2cosf)n_= X+ _)
= n + nn-1 n(n - 1) 1+
-- ~ Xn -tx~n-l~:.- Xn2-2
x 1.2 x'
n (n- 1) 1 I1 1
+ 1.2 + nx. -1 + -
1.2 Xfn-2 l1 I
n+ n (2n- 1) Xn-4 +...
= xa + nxn-2 +- 1.2
n(n- 1) 1, 1 1
+ 1.2 4 + n. -2 -(1)
1.2 Xn —4 n- x2.........
Taking together the first and last of these terms, the
second and next to last, and so on, we have
(2 cos ) (n + ) + n(x + n_2)
+ n (n -1) (-4 + 1 ) +
1.2 -- -- — f4....
But by the last article we have
n + = 2 cos nO, xn-2 + - 2 cos(n -2) 0,....
n Xn.-2