Question

If \( \cos^2(10^\circ) \cos(20^\circ) \cos(40^\circ) \cos(50^\circ) \cos(70^\circ) = \alpha + \frac{\sqrt{3}}{16} \cos(10^\circ) \), then \( 3\alpha^{-1} \) is equal to:
 

• A. 9/32
• B. 64
• C. 16
• D. 9/16

Hint:

When working with trigonometric identities, try to express complex expressions in terms of known identities to simplify and solve for unknown variables.

Answer 1

The Correct Option is A

Step 1: Use the given formula.
The equation involves the cosine of multiple angles, and the goal is to express the value of \( \alpha \). The provided equation is: \[ \cos^2(10^\circ) \cos(20^\circ) \cos(40^\circ) \cos(50^\circ) \cos(70^\circ) = \alpha + \frac{\sqrt{3}}{16} \cos(10^\circ). \]

Step 2: Solve for \( \alpha \).
After simplifying the equation and solving for \( \alpha \), we find that \( 3\alpha^{-1} \) is equal to \( \frac{9}{32} \).

Step 3: Conclusion.
Thus, the correct answer is (a) \( \frac{9}{32} \).