1. **State the problem:** Find the numerical value of $$\cot^2 30^\circ - \tan^2 45^\circ$$. 2. **Recall formulas and values:** - $$\cot 30^\circ = \frac{1}{\tan 30^\circ}$$ - $$\tan 30^\circ = \frac{1}{\sqrt{3}}$$ - $$\tan 45^\circ = 1$$ 3. **Calculate each term:** - $$\cot^2 30^\circ = \left(\frac{1}{\tan 30^\circ}\right)^2 = \left(\frac{1}{\frac{1}{\sqrt{3}}}\right)^2 = (\sqrt{3})^2 = 3$$ - $$\tan^2 45^\circ = 1^2 = 1$$ 4. **Subtract the values:** $$\cot^2 30^\circ - \tan^2 45^\circ = 3 - 1 = 2$$ **Final answer:** $$2$$