Plane trigonometry, by S.L. Loney.

348 TRIGONOMETRY.
when it will be found that all the terms on the right-hand
side will reduce to 1 _- 2.
Another proof will be found in Art. 358.
Equating coefficients of xn on the two sides of (1), we
have
2 cos nO = coefficient of xn in (1 - x2) [1 - 2x cos 0 + X2]-1
= coefficient of x - coefficient of x"-2 in
[1 - (2 cos - )]= coefficient of x - coefficient of xn-2 in
1 + (2 cos 0 - ) + 2(2 cos 0- )2+...... + x1-2 (2 cos 0 - )n-2 + xan-1 (2 cos 0 - x)n-l
+ xn (2 cos 0 - )n + x+1 (2 cos 0- x)n+l +...
Picking out the required coefficients as in the last
article, starting with the term
xn (2 cos 0 - x)n,
we have 2 cos nO
= (2 cos )n- (n - 1) (2 cos O)n-2 + ( 12)(n- ) (2cos 0)n4
_ (- 3)(n -4) (n- 5) (2 cos O)n- +
1.23......
- [(2 cos )n- -(n - 3) (2 cos O)n-4
(n-4) (n -5) (2 cos )- -.....]
=(2cos O)-n(2cos 0)n-2+ (~-2) +(n-3)(-)](2cos t)n-4
-3)( 3 -5) ) (2cos+1.2