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Problem 1:

Find the value of $\cos \frac{5\pi}{3}$.

Solution: $\cos \frac{5\pi}{3} = \cos \left( 2\pi - \frac{\pi}{3} \right) = \cos \frac{\pi}{3} = \frac{1}{2}$

Answer: $\text{B) } \frac{1}{2}$

Problem 2:

Find the value of $\sin 13^\circ \cos 167^\circ + \sin 77^\circ \sin 167^\circ$.

Using the formula: $\sin A \cos B + \cos A \sin B = \sin(A + B)$ $\sin 13^\circ \cos 167^\circ + \sin 77^\circ \sin 167^\circ = \sin(13^\circ + 167^\circ) = \sin 180^\circ = 0$

Answer: $\text{B) 0}$

Problem 3:

Simplify $\cot(\alpha - 120^\circ)$.

Using trigonometric identities: $\cot(\alpha - 120^\circ) = \frac{\cot \alpha + \sqrt{3}}{1 - \sqrt{3}\cot \alpha}$

Answer: $\text{D) } \frac{\sqrt{3} \cot \alpha + 1}{\cot \alpha + \sqrt{3}}$

Problem 4:

Find the value of