Answer
$\dfrac{\sin x+\sin y}{\cos x+\cos y}=\tan (\dfrac{x+y}{2})$
Work Step by Step
Consider LHS: $\dfrac{\sin x+\sin y}{\cos x+\cos y}=\dfrac{2\sin (\dfrac{x+y}{2}) \cos (\dfrac{x-y}{2})}{2\cos (\dfrac{x+y}{2}) \cos (\dfrac{x-y}{2})}$
or, $\dfrac{2\sin (\dfrac{x+y}{2}) \cos (\dfrac{x-y}{2})}{2\cos (\dfrac{x+y}{2}) \cos (\dfrac{x-y}{2})}=\dfrac{2\sin (\dfrac{x+y}{2})}{2\cos (\dfrac{x+y}{2})}$
or, $\dfrac{2\sin (\dfrac{x+y}{2})}{2\cos (\dfrac{x+y}{2})}=\tan (\dfrac{x+y}{2})$
Thus, $\dfrac{\sin x+\sin y}{\cos x+\cos y}=\tan (\dfrac{x+y}{2})$
Hence, the result has been proved.