I got this problem but its unrelated to any material that we worked on. Can some please help me to solve it and please explain it. Thanks

Question

Problem states:
Given:
x+(1/x)= square root of 3

PROVE:
(x^13)+(1/(x^13))= square root of 3

Answer 1

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)

Answer 2

solve for x in the given equation using the quadratic equation. Then, describe it on the complex plane, a vector at 60deg.

But x^13 is the same value, rotated around the plane twice. 13*60=60 +n360

QED