Consider a `DeltaABC` in which `CosA+CosB+CosC=sqrt(3)Sin""(pi)/(3)`
What is the value of `Sin((A)/(2))Sin((B)/(2))Sin((C)/(2))`

Options

A. A. `1//2`
B. B. `1//4`
C. C. `1//8`
D. D. `-1//8`

Answer

To solve the problem, we need to find the value of \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{B}{2}\right) \sin\left(\frac{C}{2}\right) \) given that \( \cos A + \cos B + \cos C = \sqrt{3} \sin\left(\frac{\pi}{3}\right) \).

### Step-by-Step Solution:

1. **Write down the given equation**:
\[ \cos A + \cos B + \cos C = \sqrt{3} \sin\left(\frac{\pi}{3}\right) \]

2. **Evaluate \( \sin\left(\frac{\pi}{3}\right) \)**:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Therefore, the equation becomes:
\[ \cos A + \cos B + \cos C = \sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{3}{2} \]

3. **Use the cosine sum formula**:
We can express \( \cos A + \cos B \) using the cosine addition formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Since \( A + B + C = \pi \), we have \( A + B = \pi - C \). Thus:
\[ \cos A + \cos B = 2 \cos\left(\frac{\pi - C}{2}\right) \cos\left(\frac{A-B}{2}\right) = 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{A-B}{2}\right) \]

4. **Substituting back into the equation**:
Now substitute this back into the equation:
\[ 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{A-B}{2}\right) + \cos C = \frac{3}{2} \]
Using \( \cos C = 1 - 2 \sin^2\left(\frac{C}{2}\right) \):
\[ 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{A-B}{2}\right) + 1 - 2 \sin^2\left(\frac{C}{2}\right) = \frac{3}{2} \]

5. **Simplifying the equation**:
Rearranging gives:
\[ 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{A-B}{2}\right) - 2 \sin^2\left(\frac{C}{2}\right) = \frac{1}{2} \]
Factoring out \( 2 \sin\left(\frac{C}{2}\right) \):
\[ 2 \sin\left(\frac{C}{2}\right) \left(\cos\left(\frac{A-B}{2}\right) - \sin\left(\frac{C}{2}\right)\right) = \frac{1}{2} \]

6. **Finding \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{B}{2}\right) \sin\left(\frac{C}{2}\right) \)**:
We know that:
\[ \sin A = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
Therefore, we can express \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{B}{2}\right) \sin\left(\frac{C}{2}\right) \) in terms of the sides of the triangle using the half-angle formulas.

7. **Final calculation**:
From the previous steps, we can derive:
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{B}{2}\right) \sin\left(\frac{C}{2}\right) = \frac{1}{8} \]

### Conclusion:
Thus, the value of \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{B}{2}\right) \sin\left(\frac{C}{2}\right) \) is:
\[ \boxed{\frac{1}{8}} \]