In triangle ABC,`(sinA+sinB+sinC)/(sinA+sinB-sinC)` is equal to

Options

A. `tan((A)/(2))cot((B)/(2))`
B. `cot((A)/(2))tan((B)/(2))`
C. `cot((A)/(2))cot((B)/(2))`
D. `tan((A)/(2))tan((B)/(2))`

Explanation

To solve the problem, we need to simplify the expression \(\frac{\sin A + \sin B + \sin C}{\sin A + \sin B - \sin C}\) in triangle ABC.

### Step-by-Step Solution:

1. **Understanding the Triangle Properties**:
In any triangle, the angles \(A\), \(B\), and \(C\) satisfy the equation \(A + B + C = \pi\). This means we can express \(C\) as \(C = \pi - A - B\).

2. **Using the Sine Rule**:
The sine of angle \(C\) can be rewritten using the sine addition formula:
\[ \sin C = \sin(\pi - A - B) = \sin(A + B) = \sin A \cos B + \cos A \sin B \]
However, for our simplification, we will directly use the sine values.

3. **Simplifying the Denominator**:
The denominator is \(\sin A + \sin B - \sin C\). We can rewrite \(\sin C\) as:
\[ \sin C = \sin A + \sin B - \sin C \]
Thus, we can express the denominator as:
\[ \sin A + \sin B - (\sin A + \sin B - \sin C) = \sin C \]

4. **Simplifying the Numerator**:
The numerator is \(\sin A + \sin B + \sin C\). Using the sine addition formula again, we can express this as:
\[ \sin A + \sin B + \sin C = \sin A + \sin B + (\sin A + \sin B - \sin C) \]
This simplifies to:
\[ 2(\sin A + \sin B) \]

5. **Final Expression**:
Now substituting back into our original expression:
\[ \frac{\sin A + \sin B + \sin C}{\sin A + \sin B - \sin C} = \frac{2(\sin A + \sin B)}{\sin C} \]

6. **Using Trigonometric Identities**:
We can use the identity for sine in terms of half angles:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Since \(A + B = \pi - C\), we can substitute:
\[ \sin C = 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{C}{2}\right) \]

7. **Conclusion**:
Therefore, the final expression simplifies to:
\[ \frac{2 \cdot 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)}{2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{C}{2}\right)} = 4 \cdot \frac{\sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)}{\sin\left(\frac{C}{2}\right) \cos\left(\frac{C}{2}\right)} \]

### Final Answer:
Thus, the value of \(\frac{\sin A + \sin B + \sin C}{\sin A + \sin B - \sin C}\) is equal to \(4 \cdot \cos\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right)\).