Plane trigonometry, by S.L. Loney.

SIN IN ASCENDING POWERS OF COS 0. 351
SIN 0
Picking out the required coefficients, we have
sinn0 _s n
+si (- 1 ) 2 *. 2- cos ))
__(_ (2 I +)(2 (2 )(-2 )3
+. + ( 2 )2(2. - )2 (-2cos)5
1.2.3..s
+........... + (2 cos s)n-l.
Hence, finally, when n-is even, we have
( 2+11Sil 2co ~2
sin 0
(- (o2fie 22) of (n2c22) (1 '_ 2in
=n cos 0- (- 2 cos 0 + n (-.2) ( — cos5
-......+ (- 1)2 +1 (2 cos 0)n-1....... (3).
N.B. It will be noted that equations (2) and (3) of this article are
simply the series of Art. 293 written backwards. This is;clear from the
method of proof, or the statement could be easily verified independently.
~296. To expand cos nO in a series of ascending
powers of cos 0.
As in Art. 294, we have
2 cos nO = coefficient of xn - coefficient of xn-2 in
(1 - 2x cos 0 + x2)-1
= coefficient of x'" - coefficient of X"-2 in
1 - (- 2 cos ) + X (X - 2 cos 0)2 -......
+- (- 1) x (x - 2 cos )r+...... (1),
as in Art.-295.