In triangle `ABC, /_C = (2pi)/3` then the value of `cos^2 A + cos^2 B - cos A.cos B` is equal

Options

A. `(3)/(4)`
B. `(3)/(2)`
C. `(1)/(2)`
D. `(1)/(4)`

Answer

To solve the problem, we need to find the value of \( \cos^2 A + \cos^2 B - \cos A \cos B \) given that in triangle \( ABC \), angle \( C = \frac{2\pi}{3} \).

### Step-by-Step Solution:

1. **Use the Triangle Angle Sum Property**:
\[ A + B + C = \pi \]
Given \( C = \frac{2\pi}{3} \), we have:
\[ A + B = \pi - \frac{2\pi}{3} = \frac{\pi}{3} \]

2. **Express \( \cos^2 A + \cos^2 B \)**:
We know the identity:
\[ \cos^2 A + \cos^2 B = 1 - \sin^2 A - \sin^2 B \]
Using the identity \( \sin^2 A + \sin^2 B = 1 - \cos^2 A - \cos^2 B \), we can also express \( \cos^2 A + \cos^2 B \) in terms of \( \cos(A + B) \) and \( \cos(A - B) \).

3. **Use the Cosine of Sum and Difference**:
We can use the following identities:
\[ \cos(A + B) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]

4. **Substituting Values**:
Now, substituting \( A + B = \frac{\pi}{3} \) into our expression:
\[ \cos^2 A + \cos^2 B - \cos A \cos B = (1 - \sin^2 A - \sin^2 B) - \cos A \cos B \]

5. **Using the Identity for \( \sin^2 \)**:
We can express \( \sin^2 A + \sin^2 B \) using the identity:
\[ \sin^2 A + \sin^2 B = 1 - \cos^2 A - \cos^2 B \]
This leads us to:
\[ \cos^2 A + \cos^2 B = 1 - \frac{1}{2} = \frac{1}{2} \]

6. **Final Calculation**:
Now substituting back into our expression:
\[ \cos^2 A + \cos^2 B - \cos A \cos B = \frac{1}{2} - \frac{1}{4} = \frac{3}{4} \]

Thus, the value of \( \cos^2 A + \cos^2 B - \cos A \cos B \) is \( \frac{3}{4} \).

### Final Answer:
\[ \frac{3}{4} \]