If sin A + sin B = α and cos A + cos B = β,

Given:

sin A + sin B = α            .....(i)

cos A + cos B = β           .....(ii)

Dividing (i) by (ii):

$\Rightarrow \frac{\sin A+\sin B}{\cos A+\cos B}=\frac{\alpha}{\beta}$

$\Rightarrow \frac{2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}{2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}=\frac{\alpha}{\beta}$

$\left[\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right.$ and $\left.\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$

$\Rightarrow \frac{\sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}{\cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}=\frac{\alpha}{\beta}$

$\Rightarrow \tan \left(\frac{A+B}{2}\right)=\frac{\alpha}{\beta}$