Question

Prove that if $x+\frac{1}{x}=2cos\alpha ,then{x}^n+\frac{1}{x^n}=2cosn\alpha .$

Answer 1

$x+\frac{1}{x}=2\cos \alpha$ (Given)

To Prove:

$x^n+\frac{1}{2^{:}n}=2\cos n\alpha$
Proof:

$x^2+1=2x\cos \alpha$-----------------(1)

then,

$x^{2n}+1=2x^k\cos \left(n\theta \right)$----------------------(2)

By Induction:

$n=1:x^2+1=2\cos \theta$--------------------------This is just Equation 1 is so true.

From Pythagoras and trigonometric relations, this also means:

$\sqrt{4x^2-(x^2+1{)}^2}=2x\sin \theta =(x^2+1)\tan \theta$

Assume that, $n=k:x^{2k}+1=2x^k\cos \left(k\theta \right)$ is true.

Then,

$n=k+1:x^{2k+1}\cos (k\theta +\theta )$

Now, $x=\cos \theta +i\sin \theta$

$x^n+x^{-n}=\cos n\theta +i\sin n\theta +\cos n\theta -i\sin n\theta =2\cos n\theta$

$\therefore x^n+\frac{1}{x^n}=2\cos n\alpha$