Plane trigonometry, by S.L. Loney.

430'

TRIGONOMETRY.

The factors corresponding to the first of these pairs are
x - 1 and x + 1, i.e. the quadratic factor x2- 1.
Those corresponding to the second pair are
27.. 2-r7 2.. 27r
x-cos -i sin and x -cos -+ sin-,
n n n n
i.e. the quadratic factor
2wr
2- 2x cos -+ 1.
n
Hence we get j pairs of quadratic factors.
When multiplied together they give the correct
coefficient for x, so that no constant quantity need be
prefixed to their product.
Hence, finally, when n is even,
-l = (2 -1) (2-2xcos-+ 1) (2 2x cos-+ 1)...( - 2 os 7r + 1)............. (2).
Secondly, let n be odd.
As in Art. 271 the values of the expression (1)
are now
27. 27r 47r.. 47
cos 0 + i sin 0, cos- + sin-, cos - z sin-,...?n n n n
n-3.. n-3 n-1.. n-1... cos + i sin --, cos — r + sin 7r.
n - n, n n
The first pair reduces to the single factor x - 1.
Taking the other pairs together, as before, we obtain,
when n is odd,
x - 1-( - 1) x2x- os-+l 2xc os +1...... L2 - 2 cos r+ 1 -............ (3).. n }