If A - B = `(pi)/(3)`, then find cos A cos B + sin A sin B.

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. 2
D. 1

Answer

To solve the problem, we need to find the value of \( \cos A \cos B + \sin A \sin B \) given that \( A - B = \frac{\pi}{3} \).

### Step-by-Step Solution:

1. **Identify the Trigonometric Identity**:
We know from trigonometric identities that:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
This means that \( \cos A \cos B + \sin A \sin B \) can be expressed as \( \cos(A - B) \).

2. **Substitute the Given Value**:
We are given that:
\[ A - B = \frac{\pi}{3} \]
Therefore, we can substitute this into our identity:
\[ \cos(A - B) = \cos\left(\frac{\pi}{3}\right) \]

3. **Calculate \( \cos\left(\frac{\pi}{3}\right) \)**:
The value of \( \cos\left(\frac{\pi}{3}\right) \) is known:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

4. **Final Result**:
Thus, we have:
\[ \cos A \cos B + \sin A \sin B = \cos(A - B) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Conclusion:
The value of \( \cos A \cos B + \sin A \sin B \) is \( \frac{1}{2} \).