$\dfrac{{\sin x + \sin \dfrac{x}{2}}}{{1 + \cos x + \cos \dfrac{x}{2}}} = \dfrac{{2\sin \dfrac{x}{2}\cos \dfrac{x}{2} + \sin \dfrac{x}{2}}}{{2{{\cos }^2}\dfrac{x}{2} + \cos \dfrac{x}{2}}} = \dfrac{{\sin \dfrac{x}{2}\left( {2\cos \dfrac{x}{2} + 1} \right)}}{{\cos \dfrac{x}{2}\left( {2\cos \dfrac{x}{2} + 1} \right)}} = \tan \dfrac{x}{2}$