1. The problem states: Given $\cos \theta = \frac{1}{2} \left(a + \frac{1}{a}\right)$, find the value of $\cos 5\theta$. 2. We use the multiple-angle formula for cosine: $$\cos 5\theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$$ 3. Substitute $\cos \theta = \frac{1}{2} \left(a + \frac{1}{a}\right)$ into the formula. 4. Let $x = \frac{1}{2} \left(a + \frac{1}{a}\right)$. Then: $$\cos 5\theta = 16 x^5 - 20 x^3 + 5 x$$ 5. Note that $x = \frac{a + \frac{1}{a}}{2}$ is the standard form for $\cos \theta$ when $a = e^{i\theta}$ or similar. 6. Using the identity for $\cos n\theta$ in terms of $a^n + \frac{1}{a^n}$: $$\cos n\theta = \frac{1}{2} \left(a^n + \frac{1}{a^n}\right)$$ 7. Therefore, $\cos 5\theta = \frac{1}{2} \left(a^5 + \frac{1}{a^5}\right)$. 8. Comparing with the options, the correct answer is option D: $$\frac{1}{2} \left(a^5 + \frac{1}{a^5}\right)$$. Final answer: $\boxed{\frac{1}{2} \left(a^5 + \frac{1}{a^5}\right)}$