Plane trigonometry, by S.L. Loney.

360 TRIGONOMETRY.
If cos nO be now given, the equations (1) and (2) give cos 0.
But since cos nO = cos (nO + 27r) = cos (nO + 47r)
these equations would also give
cos (0+2n, cos 0+4,...
Hence, in each case, the roots are
cos0, cos (O0+ 2), cos (0+4f )...... ton terms.
In (1) and (2) put c=- and multiply by yn.
y
We have then the equations
(-1) 2 cos n xynn. yn-l+ n ( n - )yn-3-...0......(3),
when n is odd,
n2
and [(_1)2 cos n-l yn2 y-2-.................(4),
when n is even.
The roots of these equations are respectively
secO, se(+ sec), sec + 4),....
Call these Y1, Y2,..., Yn.
Then
Y1 + Y2 +2+ + Yn= sum of the roots
n-l
- n- = (-1) 2 n sec nO, when n is odd,
(-1) 2 cosn0
and =0, when n is even.
Also.yi2 + y22 +... + y 2= (Yl + Y2 +... + yn)2 _- 2 (y Y2 + Y2Y3 +...)
n2
- -=n 2 sec2 nO, when n is odd,
cos2 nO
n2
and - 2. n - when n is even.
-1 cos nO- 1 1-(-1) cos n