If `sinalpha+sinbeta=a` and `cosalpha+cosbeta=b ,` prove that `tan((alpha-beta)/2)=+-sqrt((4-a^2-b^2)/(a^2+b^2))` .

Explanation

To prove that \( \tan\left(\frac{\alpha - \beta}{2}\right) = \pm \sqrt{\frac{4 - a^2 - b^2}{a^2 + b^2}} \) given that \( \sin \alpha + \sin \beta = a \) and \( \cos \alpha + \cos \beta = b \), we can follow these steps:

### Step 1: Square Both Sides
Start by squaring both equations:
\[ (\sin \alpha + \sin \beta)^2 = a^2 \]
\[
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