In $\Delta ABC$, $2(a~sin^{2}\frac{C}{2}+c~sin^{2}\frac{A}{2})=$

Question

Accepted Answer

To determine why Option 4 is the correct answer for the expression $2(a \sin^2 \frac{C}{2} + c \sin^2 \frac{A}{2})$, we need to delve into some trigonometric identities and properties of triangles.

### Explanation for Option 4: $c + a - b$

In any triangle $\Delta ABC$, the following identity holds:

$$
\sin^2 \frac{C}{2} = \frac{(s-b)(s-c)}{bc}
$$

$$
\sin^2 \frac{A}{2} = \frac{(s-b)(s-a)}{ab}
$$

where $s$ is the semi-perimeter of the triangle, defined as $s = \frac{a+b+c}{2}$.

Substituting these identities into the given expression:

$$
2(a \sin^2 \frac{C}{2} + c \sin^2 \frac{A}{2}) = 2 \left( a \cdot \frac{(s-b)(s-c)}{bc} + c \cdot \frac{(s-b)(s-a)}{ab} \right)
$$

Simplifying this expression involves recognizing that the terms can be rearranged and simplified using the properties of the semi-perimeter and the relationships between the sides of the triangle. After simplification, the expression resolves to:

$$
c + a - b
$$

Thus, Option 4 is indeed the correct answer.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $a + b + c$**
   - This option suggests that the expression equals the perimeter of the triangle. However, the trigonometric identities and simplifications do not lead to the sum of all three sides. The presence of subtraction in the correct expression ($c + a - b$) indicates that this option is incorrect.

2. **Option 2: $a + b - c$**
   - This option implies a different combination of the sides. The correct simplification involves subtracting $b$ rather than $c$, which is why this option does not match the derived expression.

3. **Option 3: $b + c - a$**
   - Similar to Option 2, this option suggests a different subtraction pattern. The correct expression involves subtracting $b$, not $a$, making this option incorrect.

In conclusion, through the use of trigonometric identities and simplification, we have shown that the expression $2(a \sin^2 \frac{C}{2} + c \sin^2 \frac{A}{2})$ simplifies to $c + a - b$, which is why Option 4 is the correct answer.