Question

If $\frac{1}{x}+x=2cos\theta ,$ then $x^n+\frac{1}{x^n}$ is equal to

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

Answer 1

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

Given

$x+\frac{1}{x}=2\cos \theta$

$⟹x^2+1=2x\cos \theta$

$⟹x^2-2x\cos \theta +1=0$

By Quadratic formula

$⟹x=\frac{2\cos \theta \pm \sqrt{4{\cos }^2\theta -4}}{2}$

$⟹x=\cos \theta \pm i\sin \theta$

$⟹x=e^{\pm i\theta }$

$(\cos \theta +i\sin \theta =e^{i\theta })$

Consider

$x^n+\frac{1}{x^n}$

$⟹e^{\pm in\theta }+\frac{1}{e^{\pm in\theta }}$

$⟹e^{\pm in\theta }+e^{\mp in\theta }$

$⟹2\cos n\theta$