1. **State the problem:** Marion wants to find $\cos \frac{\pi}{12}$ given that $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$.\n\n2. **Recall the formula:** We can use the half-angle formula for cosine: $$\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}.$$\n\n3. **Apply the formula:** Here, $\theta = \frac{\pi}{6}$, so $$\cos \frac{\pi}{12} = \cos \frac{\pi/6}{2} = \sqrt{\frac{1 + \cos \frac{\pi}{6}}{2}}.$$\n\n4. **Substitute the known value:** $$\cos \frac{\pi}{12} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}.$$\n\n5. **Simplify the expression:** $$\cos \frac{\pi}{12} = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}}.$$\n\n6. **Simplify the square root:** $$\cos \frac{\pi}{12} = \frac{\sqrt{2 + \sqrt{3}}}{2}.$$\n\n7. **Determine the sign:** Since $\frac{\pi}{12}$ is in the first quadrant, cosine is positive, so the positive root is taken.\n\n**Final answer:** $$\cos \frac{\pi}{12} = \frac{\sqrt{2 + \sqrt{3}}}{2}.$$