If `cos (A+B) = alpha cos A cos B + beta sin A sin B,` then `(alpha, beta)` is

Options

A. `(-1, -1)`
B. `(-1, 1)`
C. `(1,-1)`
D. `(1,1)`

Answer

To solve the problem, we need to compare the given expression for \( \cos(A + B) \) with the standard trigonometric identity for \( \cos(A + B) \).

### Step-by-Step Solution:

1. **Write down the given expression:**
\[ \cos(A + B) = \alpha \cos A \cos B + \beta \sin A \sin B \]

2. **Recall the standard formula for \( \cos(A + B) \):**
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

3. **Compare the two expressions:**
From the standard formula, we can see that:
\[ \cos(A + B) = 1 \cdot \cos A \cos B - 1 \cdot \sin A \sin B \]
This means that:
- The coefficient of \( \cos A \cos B \) is \( 1 \).
- The coefficient of \( \sin A \sin B \) is \( -1 \).

4. **Identify the values of \( \alpha \) and \( \beta \):**
From the comparison, we have:
\[ \alpha = 1 \quad \text{and} \quad \beta = -1 \]

5. **Write the final answer:**
Therefore, the ordered pair \( (\alpha, \beta) \) is:
\[ (\alpha, \beta) = (1, -1) \]

### Final Answer:
\[ (\alpha, \beta) = (1, -1) \]