If `sinA+sinB=a `and `cosA+cosB=b` then prove that `sin(A+B)=(2ab)/(a^2+b^2)` and `cos(A+B)=(b^2-a^2)/(a^2+b^2)`

Explanation

To prove the identities \( \sin(A+B) = \frac{2ab}{a^2 + b^2} \) and \( \cos(A+B) = \frac{b^2 - a^2}{a^2 + b^2} \) given that \( \sin A + \sin B = a \) and \( \cos A + \cos B = b \), we can follow these steps:

### Step 1: Use the sum-to-product identities
We start with the given equations:
1. \( \sin A + \sin B = a \)
2. \( \cos A + \cos B = b \)

Using the sum-to-product identities, we can rewrite these equations:
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