Plane trigonometry, by S.L. Loney.

FACTORS OF X2n - 2xn COS nO + 1.

425

Hence
x2n - 2xn cos nO + 1
= x2 - 2x cos 0+ 1 {x2-2xcos 0 + n) +1}
{x2-2xcos (0+n) +1}... {x2_ 2x cos (0+ n r)+L}.... (l).
By dividing by xn we have
x +- - 2cosnO= {+- 2cos }{+ —2cos 0+ -)
X x n... + — 2 cos (0 + 2 - 7r)..........(2).
The relation (2) may be written
I r=n -l f I / 2rw77\r
xXn -2 cos n= rI x + —2 cos(0+ -)
Xn r=O X,
':=n -1
where II stands for the product for all integral values
r=O
of r from r 0 to r= n - 1 of the expression following it.
Similarly we may shew that
X2' - 2an n cos nO + a2n
= {2-2at cosO+ a2} {x2 a 2ax cos ( + ) +a
{x22axcos (0+ ) +a2l. {~2axcos (0- 7) +a........... (3).
363. The proposition of the last article may also be proved by
induction.