If `A+B=C,` then
`cos^(2)A+cos^(2)B+cos^(2)C-2cosAcosBcosC=`

Options

A. 1
B. 2
C. 0
D. 3

Answer

To solve the problem, we need to evaluate the expression \( \cos^2 A + \cos^2 B + \cos^2 C - 2 \cos A \cos B \cos C \) given that \( A + B = C \).

### Step-by-step Solution:

1. **Use the identity for \( C \)**: Since \( C = A + B \), we can express \( \cos C \) using the cosine addition formula:
\[ \cos C = \cos(A + B) = \cos A \cos B - \sin A \sin B \]

2. **Express \( \cos^2 C \)**: Using the identity \( \cos^2 C = (\cos A \cos B - \sin A \sin B)^2 \):
\[ \cos^2 C = \cos^2 A \cos^2 B - 2 \cos A \cos B \sin A \sin B + \sin^2 A \sin^2 B \]

3. **Substituting \( \sin^2 A \) and \( \sin^2 B \)**: We know that \( \sin^2 A = 1 - \cos^2 A \) and \( \sin^2 B = 1 - \cos^2 B \). Substitute these into the equation:
\[ \cos^2 C = \cos^2 A \cos^2 B - 2 \cos A \cos B \sqrt{(1 - \cos^2 A)(1 - \cos^2 B)} + (1 - \cos^2 A)(1 - \cos^2 B) \]

4. **Combine terms**: Now we can combine all the terms in the expression:
\[ \cos^2 A + \cos^2 B + \cos^2 C - 2 \cos A \cos B \cos C \]

5. **Substituting \( \cos C \)**: Replace \( \cos C \) with \( \cos A \cos B - \sin A \sin B \) in the expression:
\[ \cos^2 A + \cos^2 B + \cos^2 C - 2 \cos A \cos B (\cos A \cos B - \sin A \sin B) \]

6. **Simplifying the expression**: After substituting and simplifying, we find that the expression reduces to:
\[ 1 \]

Thus, the final result is:
\[ \cos^2 A + \cos^2 B + \cos^2 C - 2 \cos A \cos B \cos C = 1 \]

### Conclusion:
The value of the expression is \( 1 \).