Question

\( \cos(36^\circ - A) \cos(36^\circ + A) + \cos(54^\circ + A) \cos(54^\circ - A) = \)

• A. \( \cos 2A \)
• B. \( \cos A \)
• C. \( \sin 2A \)
• D. \( \sin A \)

Hint:

To simplify expressions with products of trigonometric functions, use the product-to-sum or sum-to-product identities.

Answer 1

The Correct Option is A

Step 1: Use the product-to-sum identities.
We use the trigonometric identity for the product of cosines: \[ \cos x \cos y = \frac{1}{2} \left( \cos(x - y) + \cos(x + y) \right) \] By applying this identity to the given expression, we simplify the terms and find that the result is \( \cos 2A \).

Step 2: Conclusion.
Thus, the correct answer is \( \cos 2A \), corresponding to option (A).