Question

If \( A = \frac{\pi}{24} \), then \[ \frac{\cos A + \cos 3A + \cos 5A + \cos 7A}{\sin A + \sin 3A + \sin 5A + \sin 7A} \]

• A. \( \sqrt{3} \)
• B. \( 2\sqrt{3} \)
• C. \( \frac{1}{\sqrt{3}} \)
• D. \( \frac{2}{\sqrt{3}} \)

Hint:

Use sum-to-product identities to simplify trigonometric expressions.

Answer 1

The Correct Option is A

We apply the sum-to-product identities to the given trigonometric expression: \[ \cos A + \cos 7A = 2 \cos 4A \cos 3A, \quad \cos 3A + \cos 5A = 2 \cos 4A \cos A \] \[ \sin A + \sin 7A = 2 \sin 4A \cos 3A, \quad \sin 3A + \sin 5A = 2 \sin 4A \cos A \] The expression simplifies to: \[ \frac{2 \cos 4A (2 \cos 2A \cos A)}{2 \sin 4A (2 \cos 2A \cos A)} = \frac{\cos 4A}{\sin 4A} = \cot 4A \] Since \( 4A = \frac{\pi}{6} \), we have: \[ \cot \left( \frac{\pi}{6} \right) = \sqrt{3} \] Thus, the answer is \( \sqrt{3} \), option (1).