In a `DeltaABC` , which one of the following is true ?

Options

A. `(b+c)"cos"(A)/(2)=asin((B+C)/(2))`
B. `(b+c)cos((B+C)/(2))=a"sin"(A)/(2)`
C. `(b-c)cos((B-C)/(2))=a"cos"(A)/(2)`
D. `(b-c)"cos"(A)/(2)=asin((B-C)/(2))`

Answer

To determine which of the given options is true for triangle ABC, we will analyze each option step by step.

### Step-by-Step Solution

1. **Understanding the Triangle Properties**:
In triangle ABC, we know that the sum of the angles is \( A + B + C = \pi \) (or \( 180^\circ \)).

2. **Analyzing Option 1**:
- The first option states:
\[ b + c \cos\left(\frac{A}{2}\right) = a \sin\left(\frac{B + C}{2}\right) \]
- Since \( B + C = \pi - A \), we have:
\[ \sin\left(\frac{B + C}{2}\right) = \sin\left(\frac{\pi - A}{2}\right) = \cos\left(\frac{A}{2}\right) \]
- Thus, the equation simplifies to:
\[ b + c \cos\left(\frac{A}{2}\right) = a \cos\left(\frac{A}{2}\right) \]
- Rearranging gives:
\[ b + c = a \]
- This is not necessarily true for all triangles, hence **Option 1 is incorrect**.

3. **Analyzing Option 2**:
- The second option states:
\[ b + c \cos\left(\frac{B + C}{2}\right) = a \sin\left(\frac{A}{2}\right) \]
- Using \( B + C = \pi - A \):
\[ \cos\left(\frac{B + C}{2}\right) = \sin\left(\frac{A}{2}\right) \]
- Thus, the equation becomes:
\[ b + c \sin\left(\frac{A}{2}\right) = a \sin\left(\frac{A}{2}\right) \]
- Rearranging gives:
\[ b + c = a \]
- This is also not necessarily true for all triangles, hence **Option 2 is incorrect**.

4. **Analyzing Option 3**:
- The third option states:
\[ b - c \cos\left(\frac{B - C}{2}\right) = a \cos\left(\frac{A}{2}\right) \]
- We can test this with an equilateral triangle where \( a = b = c \).
- Substituting values leads to:
\[ b - c \cdot 1 = a \cdot \frac{1}{2} \]
- This does not hold true in general, hence **Option 3 is incorrect**.

5. **Analyzing Option 4**:
- The fourth option states:
\[ b - c \cos\left(\frac{A}{2}\right) = a \sin\left(\frac{B - C}{2}\right) \]
- We can use the half-angle identities and properties of triangles to analyze this.
- Testing with an equilateral triangle where \( a = b = c \):
\[ 0 = 0 \]
- This holds true, and further analysis shows that it can be derived from the properties of triangles, hence **Option 4 is correct**.

### Conclusion
After analyzing all options, we conclude that **Option 4 is the correct statement for triangle ABC**.