Question

If \( f(x) = \cos x \), find the following expression: \[ \frac{1}{2} \left[ f(x + y) + f(y - x) - f(x) \cdot f(y) \right] \]

• A. \( \cos(x + y) + \cos(y - x) - \cos x \cdot \cos y \)
• B. \( \cos(x + y) - \cos(y - x) - \cos x \cdot \cos y \)
• C. \( 2 \cos(x + y) - \cos x \cdot \cos y \)
• D. \( \cos(x + y) + \cos(y - x) + \cos x \cdot \cos y \)

Hint:

When dealing with trigonometric identities, remember that cosine is an even function, meaning \( \cos(x - y) = \cos(y - x) \). This can help simplify expressions involving cosine.

Answer 1

The Correct Option is A

We are given that \( f(x) = \cos x \). Therefore, we need to evaluate the following expression: \[ \frac{1}{2} \left[ f(x + y) + f(y - x) - f(x) \cdot f(y) \right] \] Substitute \( f(x) = \cos x \) into the expression: \[ = \frac{1}{2} \left[ \cos(x + y) + \cos(y - x) - \cos x \cdot \cos y \right] \]

1. Step 1: Simplify the expression: We use the property that \( \cos(y - x) = \cos(x - y) \) (since cosine is an even function). Thus, the expression simplifies to: \[ = \frac{1}{2} \left[ \cos(x + y) + \cos(x - y) - \cos x \cdot \cos y \right] \]

2. Step 2: Final answer: The final result is: \[ \cos(x + y) + \cos(y - x) - \cos x \cdot \cos y \] which corresponds to option (A).