1. **Problem statement:** Given $\cos A = \frac{\sqrt{3}}{2}$, find the value of $\cos 2A$. 2. **Formula used:** The double-angle formula for cosine is: $$\cos 2A = 2\cos^2 A - 1$$ This formula allows us to find the cosine of twice an angle if we know the cosine of the original angle. 3. **Substitute the given value:** $$\cos 2A = 2\left(\frac{\sqrt{3}}{2}\right)^2 - 1$$ 4. **Simplify inside the parentheses:** $$\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$$ 5. **Calculate:** $$\cos 2A = 2 \times \frac{3}{4} - 1 = \frac{6}{4} - 1 = \frac{3}{2} - 1 = \frac{1}{2}$$ 6. **Final answer:** $$\cos 2A = \frac{1}{2}$$