If `A + B =(pi)/(3)` and cosA + cosB = 1then the value of cos`((A-B)/2)`is_______

Answer

To solve the problem, we are given that \( A + B = \frac{\pi}{3} \) and \( \cos A + \cos B = 1 \). We need to find the value of \( \cos\left(\frac{A-B}{2}\right) \).

### Step-by-step Solution:

1. **Use the Cosine Sum Formula**:
We know from trigonometric identities that:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Given \( A + B = \frac{\pi}{3} \), we can substitute this into the formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{\frac{\pi}{3}}{2}\right) \cos\left(\frac{A-B}{2}\right) \]

2. **Calculate \( \cos\left(\frac{A+B}{2}\right) \)**:
\[ \frac{A+B}{2} = \frac{\frac{\pi}{3}}{2} = \frac{\pi}{6} \]
Now, find \( \cos\left(\frac{\pi}{6}\right) \):
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

3. **Set Up the Equation**:
Now substituting back into the equation:
\[ \cos A + \cos B = 2 \cdot \frac{\sqrt{3}}{2} \cdot \cos\left(\frac{A-B}{2}\right) \]
This simplifies to:
\[ \cos A + \cos B = \sqrt{3} \cdot \cos\left(\frac{A-B}{2}\right) \]

4. **Use the Given Condition**:
We know from the problem statement that \( \cos A + \cos B = 1 \). Therefore, we can set up the equation:
\[ 1 = \sqrt{3} \cdot \cos\left(\frac{A-B}{2}\right) \]

5. **Solve for \( \cos\left(\frac{A-B}{2}\right) \)**:
Rearranging the equation gives:
\[ \cos\left(\frac{A-B}{2}\right) = \frac{1}{\sqrt{3}} \]

6. **Final Result**:
Thus, the value of \( \cos\left(\frac{A-B}{2}\right) \) is:
\[ \cos\left(\frac{A-B}{2}\right) = \frac{1}{\sqrt{3}} \]

### Final Answer:
The value of \( \cos\left(\frac{A-B}{2}\right) \) is \( \frac{1}{\sqrt{3}} \).