1. **Problem Statement:** Find $\sec 30^\circ$ given that $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\cot 30^\circ = \sqrt{3}$.\n\n2. **Recall the definition:** $\sec \theta = \frac{1}{\cos \theta}$. This means the secant of an angle is the reciprocal of the cosine of that angle.\n\n3. **Apply the formula:** Since $\cos 30^\circ = \frac{\sqrt{3}}{2}$, then\n$$\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}}.$$\n\n4. **Simplify the expression:**\n$$\sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}.$$\n\n5. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{3}$ to get rid of the square root in the denominator:\n$$\sec 30^\circ = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}.$$\n\n6. **Final answer:**\n$$\boxed{\sec 30^\circ = \frac{2\sqrt{3}}{3}}.$$\n\n**Multiple-choice options:**\n1. $\frac{\sqrt{3}}{3}$\n2. $\frac{\sqrt{2}}{2}$\n3. $2 \times \frac{\sqrt{3}}{3}$\n4. $2 \times \frac{\sqrt{2}}{2}$\n5. None of these\n6. $\frac{1}{2}$\n7. $1$\n\nThe correct choice is option 3: $2 \times \frac{\sqrt{3}}{3}$.