Prove that: `(cosA+cosB)^2+(sinA-sinB)^2=4cos^2((A+B)/2)` .

Explanation

To prove the identity \((\cos A + \cos B)^2 + (\sin A - \sin B)^2 = 4 \cos^2\left(\frac{A+B}{2}\right)\), we will start from the left-hand side and manipulate it step by step.

### Step 1: Expand the left-hand side
We start with the left-hand side:
\[ (\cos A + \cos B)^2 + (\sin A - \sin B)^2 \]

...