Plane trigonometry, by S.L. Loney.

432 TRIGONOMETRY.
The quadratic factor corresponding to the second
pair is
37r
x2- 2x cos - + 1,
and so on.
Hence, as in the last article, when n is even, we have
xn -+1= x(2- 2x cos + 1) (x 2x cos -- )...
\ n )\ n.. x2- 2x cos ( +.
n
Secondly, let n be odd.
The values of the expression (1) are in this case
7r.. 3.. 3
cos - +sin-, cos + i sin....
cos + isin, cos- + sin-.
n n n n
The last pair of roots reduces to the single root - 1, so
that x + 1 is one of the required factors.
The quadratic factors corresponding to the successive
pairs of roots are
37r
x-2xcos7-+1, x2-2xcos +1,...
n —2
X2 - 2x cos - r + 1.
n
Hence finally, when n is odd, we have
n+ 1 =($~+ 1) (2 _- 2x cos-+)+ (2 - 2X cos -....[W os(- +1]n... 2- 2xcos(n - + 1.
n