1. Stating the problem: Calculate the values of the following expressions involving trigonometric functions: a. $\sin 60^\circ \cos 45^\circ + \sec 30^\circ \cot 60^\circ$ b. $\frac{\sin 30^\circ + \cos^2 45^\circ + \tan^2 60^\circ}{\csc 30^\circ \cot 45^\circ}$ 2. Solve part (a): - Recall values: $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{2}{\sqrt{3}}$, $\cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}}$ - Compute each term: $\sin 60^\circ \cos 45^\circ = \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}$ - Compute $\sec 30^\circ \cot 60^\circ = \frac{2}{\sqrt{3}} \times \frac{1}{\sqrt{3}} = \frac{2}{3}$ - Sum: $\frac{\sqrt{6}}{4} + \frac{2}{3} = \frac{3\sqrt{6}}{12} + \frac{8}{12} = \frac{3\sqrt{6} + 8}{12}$ 3. Solve part (b): - Recall values: $\sin 30^\circ = \frac{1}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$ so $\cos^2 45^\circ = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2}$ - $\tan 60^\circ = \sqrt{3}$ so $\tan^2 60^\circ = 3$ - Numerator: $\sin 30^\circ + \cos^2 45^\circ + \tan^2 60^\circ = \frac{1}{2} + \frac{1}{2} + 3 = 4$ - Denominator: $\csc 30^\circ = \frac{1}{\sin 30^\circ} = 2$, $\cot 45^\circ = 1$ - Denominator: $2 \times 1 = 2$ - Fraction: $\frac{4}{2} = 2$ Final answers: a. $\frac{3\sqrt{6} + 8}{12}$ b. $2$