Question

If $ x + \frac{1}{x} = 2 \cos \theta $, find the value of \[ x^5 + \frac{1}{x^5} \] in terms of $\theta$.

• A. $2 \cos 5\theta$
• B. $5 \cos \theta$
• C. $2 \cos 4\theta$
• D. $5 \cos 5\theta$

Hint:

Use the recurrence relation $a_n = (x + \frac{1}{x}) a_{n-1} - a_{n-2}$ and trigonometric identities to express powers in terms of cosines.