Question

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

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

Answer 1

Answer: (A)

$2\cos n\theta$

Let $x=\cos \theta +i\sin \theta \Rightarrow x^n=\cos n\theta +i\sin n\theta$
Then, $\frac{1}{x}=\cos \theta -i\sin \theta \Rightarrow \frac{1}{x^n}=\cos n\theta -i\sin n\theta$
$\therefore x^n+\frac{1}{x^n}=2\cos n\theta$