If `cos(A+B)=alphacosAcosB+betasinAsinB`, then `(alpha,beta)=?`

Options

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

Answer

To solve the equation \( \cos(A + B) = \alpha \cos A \cos B + \beta \sin A \sin B \) and find the values of \( \alpha \) and \( \beta \), we will compare it with the known trigonometric identity for \( \cos(A + B) \).

### Step-by-Step Solution:

1. **Recall the Trigonometric Identity**:
The trigonometric identity for the cosine of the sum of two angles is:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

2. **Set Up the Comparison**:
We need to compare this identity with the given expression:
\[ \cos(A + B) = \alpha \cos A \cos B + \beta \sin A \sin B \]

3. **Identify Coefficients**:
From the identity \( \cos(A + B) = \cos A \cos B - \sin A \sin B \), we can identify the coefficients:
- The coefficient of \( \cos A \cos B \) is \( 1 \).
- The coefficient of \( \sin A \sin B \) is \( -1 \).

4. **Equate Coefficients**:
By comparing the coefficients from both sides, we have:
\[ \alpha = 1 \]
\[ \beta = -1 \]

5. **Conclusion**:
Therefore, the values of \( \alpha \) and \( \beta \) are:
\[ (\alpha, \beta) = (1, -1) \]

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