Plane trigonometry, by S.L. Loney.

SIN nO
IN DESCENDING POWERS OF COS 0. 347
sIN 0
Now coefficient of
xn-1 in Xn-1 (2 cos 0 - x)- = (2 cos 0)-1,
coefficient of x-l in xf-2 (2 cos 0 - X)n-2
= coefficient of x in (2 cos 0 - )?-2
= -(n - 2) (2 cos 0)-3,
coefficient of x~-l in Xn-3 (2 cos 0 - a)-3
= coefficient of x2 in (2 cos 0 - x)n(n,- 3) (n- 4) (2 c )
1.2
and so on.
Hence, from (2) picking out in this manner all the
coefficients of x'-l, we have
sin =(2 cos 0)n-1- (n - 2) (2 cos )O -3
+ (- 3) (n - 4) 2 c )n1.2
_(n- 4)(n- 5) (n- 6)(2+......
1.2.3
n2-1
If n be odd, the last term could be proved to be (-1) 2; if n be even,
it could be shewn to,be ( - 1) (n cos 0).
x#294. To express cos nO in a series of descending
powers of cos 0.
If, be < 1, we have
1 - X2
1- — Xs = -1 + 2 cos 0 + 22 cos 20 + 2x cos 30 +..
1 - 2x cos 0 + x"... + 2xn cos n +... ad inf...... (1).
This may be shewn by multiplying both sides by
1 - 2x cos 0'+ 2,