Consider a `DeltaABC` satisfying `2aSin^(2)((C)/(2))+2cSin^(2)((A)/(2))=2a+2c-3b`
`SinA, SinB, SinC` are in

Options

A. A. GP
B. B. AP
C. C. HP
D. D. None

Answer

To solve the problem, we need to analyze the given equation and determine the relationship between \( \sin A \), \( \sin B \), and \( \sin C \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ 2a \sin^2\left(\frac{C}{2}\right) + 2c \sin^2\left(\frac{A}{2}\right) = 2a + 2c - 3b \]

2. **Use the half-angle formulas for sine**:
The half-angle formulas are:
\[ \sin\left(\frac{C}{2}\right) = \sqrt{\frac{(s-a)(s-b)}{ab}} \quad \text{and} \quad \sin\left(\frac{A}{2}\right) = \sqrt{\frac{(s-b)(s-c)}{bc}} \]
where \( s = \frac{a+b+c}{2} \) is the semi-perimeter of the triangle.

3. **Substitute the half-angle formulas into the equation**:
Substitute the expressions for \( \sin^2\left(\frac{C}{2}\right) \) and \( \sin^2\left(\frac{A}{2}\right) \) into the original equation:
\[ 2a \left(\frac{(s-a)(s-b)}{ab}\right) + 2c \left(\frac{(s-b)(s-c)}{bc}\right) = 2a + 2c - 3b \]

4. **Simplify the left-hand side**:
Multiply through by \( ab \) to eliminate the denominators:
\[ 2a(s-a)(s-b) + 2c(s-b)(s-c) = (2a + 2c - 3b)ab \]

5. **Rearranging and simplifying**:
After simplifying, we will have a relationship involving \( a \), \( b \), and \( c \). The left-hand side can be expanded and combined with the right-hand side.

6. **Use the relationship between sides and angles**:
From the Law of Sines, we know:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = r \]
where \( r \) is a constant.

7. **Express \( a \), \( b \), and \( c \) in terms of \( \sin A \), \( \sin B \), and \( \sin C \)**:
We can express:
\[ a = r \sin A, \quad b = r \sin B, \quad c = r \sin C \]

8. **Substituting back into the equation**:
Substitute these expressions into the simplified equation to relate \( \sin A \), \( \sin B \), and \( \sin C \).

9. **Establish the relationship**:
After substitution and simplification, we find that:
\[ 2 \sin B = \sin A + \sin C \]
This indicates that \( \sin A \), \( \sin B \), and \( \sin C \) are in Arithmetic Progression (AP).

### Conclusion:
Thus, we conclude that \( \sin A \), \( \sin B \), and \( \sin C \) are in Arithmetic Progression (AP).