Given:
x+(1/x)= square root of 3
PROVE:
(x^13)+(1/(x^13))= square root of 3
Put x = exp(i t)
Then:
x + 1/x = 2 cos(t)
x + 1/x = sqrt(3) ---->
cos(t) = sqrt(3)/2 ---->
t = ±pi/6 (adding a multiple of 2 pi leaves x invariant)
x^(13) + x^(-13) = 2 cos(13 t) =
2 cos(13/6 pi) = 2 cos(pi/6) = sqrt(3)
