1. **Problem Statement:** Given \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\cot 30^\circ = \sqrt{3}\), find \(\sec 30^\circ\) and \(\tan 30^\circ\). 2. **Recall the definitions:** - \(\sec \theta = \frac{1}{\cos \theta}\) - \(\tan \theta = \frac{1}{\cot \theta}\) 3. **Calculate \(\sec 30^\circ\):** \[ \sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \] 4. **Calculate \(\tan 30^\circ\):** \[ \tan 30^\circ = \frac{1}{\cot 30^\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \] 5. **Interpretation:** - \(\sec 30^\circ = \frac{2\sqrt{3}}{3}\) matches option 3. - \(\tan 30^\circ = \frac{\sqrt{3}}{3}\) is also found. **Final answers:** - \(\sec 30^\circ = \frac{2\sqrt{3}}{3}\) - \(\tan 30^\circ = \frac{\sqrt{3}}{3}\)