If `cos A, cosB and cosC` are the roots of the cubic `x^3 + ax^2 + bx + c = 0` where `A, B, C` are the anglesof a triangle then find the value of `a^2 – 2b– 2c`.

Options

A. 0
B. `1//2`
C. 1
D. 2

Explanation

To solve the problem, we need to find the value of \( a^2 - 2b - 2c \) given that \( \cos A, \cos B, \cos C \) are the roots of the cubic equation \( x^3 + ax^2 + bx + c = 0 \), where \( A, B, C \) are the angles of a triangle.

### Step-by-Step Solution:

1. **Identify the roots and their relationships**:
- The roots of the cubic equation are \( \cos A, \cos B, \cos C \).
- By Vieta's formulas, we have:
- Sum of roots: \( \cos A + \cos B + \cos C = -a \)
- Product of roots: \( \cos A \cos B \cos C = -c \)
- Sum of products of roots taken two at a time: \( \cos A \cos B + \cos B \cos C + \cos C \cos A = b \)

2. **Square the sum of the roots**:
- We square the sum of the roots:
\[ (\cos A + \cos B + \cos C)^2 = (-a)^2 = a^2 \]
- Expanding the left-hand side:
\[ \cos^2 A + \cos^2 B + \cos^2 C + 2(\cos A \cos B + \cos B \cos C + \cos C \cos A) = a^2 \]
- Substituting \( \cos A \cos B + \cos B \cos C + \cos C \cos A = b \):
\[ \cos^2 A + \cos^2 B + \cos^2 C + 2b = a^2 \]

3. **Express \( \cos^2 A, \cos^2 B, \cos^2 C \)**:
- Using the identity \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \):
\[ \cos^2 A + \cos^2 B + \cos^2 C = \frac{1 + \cos 2A}{2} + \frac{1 + \cos 2B}{2} + \frac{1 + \cos 2C}{2} \]
- This simplifies to:
\[ \frac{3}{2} + \frac{\cos 2A + \cos 2B + \cos 2C}{2} \]

4. **Use the property of angles in a triangle**:
- Since \( A + B + C = \pi \), we have:
\[ \cos 2A + \cos 2B + \cos 2C = 1 - 4 \cos A \cos B \cos C \]
- Substituting \( \cos A \cos B \cos C = -c \):
\[ \cos 2A + \cos 2B + \cos 2C = 1 + 4c \]

5. **Substitute back into the equation**:
- Now substituting back into the expression for \( \cos^2 A + \cos^2 B + \cos^2 C \):
\[ \frac{3}{2} + \frac{1 + 4c}{2} = \frac{4 + 4c}{2} = 2 + 2c \]

6. **Combine everything**:
- Now substituting this into the equation we derived earlier:
\[ 2 + 2c + 2b = a^2 \]
- Rearranging gives:
\[ a^2 - 2b - 2c = 2 \]

### Final Result:
Thus, the value of \( a^2 - 2b - 2c \) is \( \boxed{2} \).