If : `A+B+C=pi, "then"" "sin ^(2) A +sin^(2)B - sin ^(2)C=` A)`2 cos A * cos B * sin C` B)`2 cos B * cos C * sin A` C)`2 sin A * sin B * cos C` D)`2 sin B * sin C * cos A`

Options

A. `2 cos A * cos B * sin C`
B. `2 cos B * cos C * sin A`
C. `2 sin A * sin B * cos C`
D. `2 sin B * sin C * cos A`

Answer

To solve the problem, we need to find the value of the expression \( \sin^2 A + \sin^2 B - \sin^2 C \) given that \( A + B + C = \pi \).

### Step-by-Step Solution:

1. **Use the identity for \( \sin^2 \)**:
We know that \( \sin^2 A = 1 - \cos^2 A \). Thus, we can rewrite the expression:
\[ \sin^2 A + \sin^2 B - \sin^2 C = (1 - \cos^2 A) + (1 - \cos^2 B) - (1 - \cos^2 C) \]
Simplifying this gives:
\[ = 2 - (\cos^2 A + \cos^2 B - \cos^2 C) \]

2. **Substituting \( C \)**:
Since \( A + B + C = \pi \), we can express \( C \) as \( C = \pi - A - B \). Using the cosine of a sum, we have:
\[ \cos C = \cos(\pi - A - B) = -\cos(A + B) \]
Therefore, \( \cos^2 C = \cos^2(A + B) \).

3. **Using the cosine addition formula**:
We can expand \( \cos(A + B) \) using the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Thus,
\[ \cos^2 C = (\cos A \cos B - \sin A \sin B)^2 \]

4. **Substituting back into the expression**:
Now we substitute back into our expression:
\[ \sin^2 A + \sin^2 B - \sin^2 C = 2 - (\cos^2 A + \cos^2 B - \cos^2(A + B)) \]

5. **Using the identity for \( \cos^2 \)**:
We can use the identity \( \cos^2 A + \cos^2 B + \cos^2 C = 1 \) (for angles summing to \( \pi \)) to relate the terms. However, we can also directly compute:
\[ \sin^2 A + \sin^2 B = 1 - \cos^2 A + 1 - \cos^2 B \]
Thus, we can express \( \sin^2 A + \sin^2 B - \sin^2 C \) in terms of \( A \) and \( B \).

6. **Final simplification**:
After substituting and simplifying, we arrive at:
\[ \sin^2 A + \sin^2 B - \sin^2 C = 2 \sin A \sin B \cos C \]

7. **Identifying the correct option**:
From the options given, we can see that the expression matches with option C:
\[ 2 \sin A \sin B \cos C \]

### Final Answer:
Thus, the answer is:
\[ \text{C) } 2 \sin A \sin B \cos C \]