$\sin \theta +\sin (\theta +120)-\sin \left({120}^{\circ }-\theta \right)=$

$\sin \left(\theta +{120}^{\circ }\right)=\sin \theta \cos {120}^{\circ }+\cos \theta \sin {120}^{\circ }$

$=-\frac{\sin \theta }{2}+\frac{\sqrt{3}\cos \theta }{2}$

$\sin \left({120}^{\circ }-\theta \right)=\sin {120}^{\circ }\cos \theta -\cos {120}^{\circ }\sin \theta$

$=\frac{\sqrt{3}\cos \theta }{2}+\frac{\sin \theta }{2}$

$\therefore \sin \theta +\sin \left(\theta +{120}^{\circ }\right)-\sin \left({120}^{\circ }-\theta \right)$

$\begin{array}{l}=\sin \theta -\frac{\sin \theta }{2}+\frac{\sqrt{3}\cos \theta }{2}-\left(\frac{\sqrt{3}\cos \theta }{2}+\frac{\sin \theta }{2}\right)\\ =\sin \theta -\frac{\sin \theta }{2}+\frac{\sqrt{3}\cos \theta }{2}-\frac{\sqrt{3}\cos \theta }{2}-\frac{\sin \theta }{2}=0\end{array}$