$\frac{\sin \theta +\sin \phi }{\cos \theta +\cos \phi }=\frac{2\sin \tfrac{\theta +\phi }{2}\cos \tfrac{\theta -\phi }{2}}{2\cos \tfrac{\theta +\phi }{2}\cos \tfrac{\theta -\theta }{2}}=\frac{\sin \tfrac{\theta +\phi }{2}}{\cos \tfrac{\theta +\phi }{2}}=\tan \left( \frac{\theta +\phi }{2} \right)$