`cos^(2)alpha+cos^(2)(alpha+120^(@))+cos^(2)(alpha-120^(@))=?`

Options

A. `(3)/(2)`
B. 1
C. `(1)/(2)`
D. 0

Answer

To solve the equation \( \cos^2 \alpha + \cos^2(\alpha + 120^\circ) + \cos^2(\alpha - 120^\circ) \), we can use a specific value for \( \alpha \) to simplify our calculations.

### Step-by-Step Solution:

1. **Choose a Value for \( \alpha \)**:
Let's set \( \alpha = 60^\circ \). This choice will simplify our calculations because we know the cosine values for common angles.

2. **Calculate Each Cosine Term**:
- First term:
\[ \cos^2(60^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
- Second term:
\[ \cos^2(60^\circ + 120^\circ) = \cos^2(180^\circ) = (-1)^2 = 1 \]
- Third term:
\[ \cos^2(60^\circ - 120^\circ) = \cos^2(-60^\circ) = \cos^2(60^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

3. **Sum the Terms**:
Now, we add all the calculated terms together:
\[ \cos^2(60^\circ) + \cos^2(180^\circ) + \cos^2(-60^\circ) = \frac{1}{4} + 1 + \frac{1}{4} \]

4. **Combine the Fractions**:
- Combine \( \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \).
- Now add \( 1 \):
\[ \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]

5. **Final Result**:
Thus, the value of \( \cos^2 \alpha + \cos^2(\alpha + 120^\circ) + \cos^2(\alpha - 120^\circ) \) is:
\[ \boxed{\frac{3}{2}} \]