For the following two (02) items: Let A + B = p and A + B = q ....

Given $p = \sin A + \sin B and$q = \cos A + \cos B$.<br><br>Using the sum-to-product formulas:<br><br>$p = 2 \sin\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right)$<br><br>$q = 2 \cos\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right)$<br><br>Dividing$p$by$q$:<br><br>$\dfrac{p}{q} = \dfrac{2 \sin\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right)}{2 \cos\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right)}$<br><br>$\dfrac{p}{q} = \tan\left(\dfrac{A+B}{2}\right)