CALCULUS

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

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

  • CALCULUS -

    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)

  • CALCULUS -

    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

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