Question

If $x+\frac{1}{x}=2\cdot \cos \theta$ then
$x^n+\frac{1}{x^n}$ equals

• A. $2\sin n\theta$
• B. $\cos n\theta$
• C. $\sin n\theta$
• D. $2\cos n\theta$

Answer 1

The correct option is D $2\cos n\theta$

$x+\frac{1}{x}=2\cos \theta$
$\Rightarrow x^2-2x\cos \theta +1=0$

$\therefore x=\cos \theta \pm i\sin \theta$

$\therefore x^n=\cos n\theta \pm i\sin n\theta ...\left(A\right)$

Also $\frac{1}{x^n}=\cos n\theta \mp i\sin n\theta ...\left(B\right)$
$\therefore x^n+\frac{1}{x^n}=2\cos n\theta$