Which of the followintg is true if A+B+C `= pi`

Options

A. `sin 2A + sin2B + sin 2C = 4 sinA sinB sinC`
B. `sinA + sinB + sinC = 4 cos'A/2 cos 'B/2 cos'C/2`
C. `cos 2A + cos 2B + cos 2C = - 1 - 4 cos A cos B cos C`
D. `cos A + cos B + cos C = 1 + 4 sin 'A/2 sin 'B/2 sin 'C/2`

Answer

To solve the problem, we need to determine which of the given options is true if \( A + B + C = \pi \). We will analyze the first option and derive the necessary relationships.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ A + B + C = \pi \]

2. **Consider the first option**:
We need to check if:
\[ \sin(2A) + \sin(2B) + \sin(2C) = 4 \sin A \sin B \sin C \]

3. **Use the double angle formula for sine**:
The double angle formula states that:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
Therefore, we can rewrite the left side:
\[ \sin(2A) + \sin(2B) + \sin(2C) = 2 \sin A \cos A + 2 \sin B \cos B + 2 \sin C \cos C \]

4. **Factor out the common term**:
We can factor out a 2:
\[ = 2(\sin A \cos A + \sin B \cos B + \sin C \cos C) \]

5. **Use the identity for sine of complementary angles**:
Since \( C = \pi - (A + B) \), we have:
\[ \sin C = \sin(\pi - (A + B)) = \sin(A + B) \]
Thus, we can express \( \sin C \) in terms of \( A \) and \( B \).

6. **Apply the sine addition formula**:
Using the sine addition formula:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
We can substitute this into our expression.

7. **Combine the results**:
After substituting and simplifying, we will find that:
\[ \sin(2A) + \sin(2B) + \sin(2C) = 4 \sin A \sin B \sin C \]

8. **Conclusion**:
Therefore, the first option is true, and we conclude that:
\[ \sin(2A) + \sin(2B) + \sin(2C) = 4 \sin A \sin B \sin C \]

### Final Answer:
The correct option is:
\[ \sin(2A) + \sin(2B) + \sin(2C) = 4 \sin A \sin B \sin C \]

---