In any ` A B C` , prove that: `2{b cos^2(C/2) + c cos^2(B/2)}=a+c+b`

Explanation

To prove the equation \( 2(b \cos^2(\frac{C}{2}) + c \cos^2(\frac{B}{2})) = a + b + c \) in triangle \( ABC \), we will follow these steps:

### Step 1: Use the Cosine Half-Angle Identity
The cosine half-angle identity states that:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2} \]
We can apply this identity to \( \cos^2\left(\frac{C}{2}\right) \) and \( \cos^2\left(\frac{B}{2}\right) \).
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