1. **State the problem:** Calculate the following trigonometric values and verify identities: - $\cot 30^\circ$ - $\sec 60^\circ$ - $\csc 90^\circ$ - $\cos^2 50^\circ + \sin^2 50^\circ$ - $\cos^2 100^\circ + \sin^2 100^\circ$ 2. **Recall formulas and identities:** - $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ - $\csc \theta = \frac{1}{\sin \theta}$ - Pythagorean identity: $\cos^2 \theta + \sin^2 \theta = 1$ 3. **Calculate each value:** - $\cot 30^\circ = \frac{\cos 30^\circ}{\sin 30^\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}$ - $\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{1/2} = 2$ - $\csc 90^\circ = \frac{1}{\sin 90^\circ} = \frac{1}{1} = 1$ - $\cos^2 50^\circ + \sin^2 50^\circ = 1$ (by Pythagorean identity) - $\cos^2 100^\circ + \sin^2 100^\circ = 1$ (by Pythagorean identity) 4. **Final answers:** - $\cot 30^\circ = \sqrt{3}$ - $\sec 60^\circ = 2$ - $\csc 90^\circ = 1$ - $\cos^2 50^\circ + \sin^2 50^\circ = 1$ - $\cos^2 100^\circ + \sin^2 100^\circ = 1$