In any ` A B C` , prove that: `2{asin^2C/2+csin^2A/2}=a+c-b`

Explanation

To prove the equation \( 2\left(a \sin^2\frac{C}{2} + c \sin^2\frac{A}{2}\right) = a + c - b \) in triangle \( ABC \), we will start from the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step-by-Step Solution:

1. **Start with the Left-Hand Side (LHS)**:
\[ \text{LHS} = 2\left(a \sin^2\frac{C}{2} + c \sin^2\frac{A}{2}\right) \]
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