$\begin{array}{l} {\sin ^2}x + {\sin ^2}\left( {\dfrac{\pi }{3} - x} \right) + \sin x\sin \left( {\dfrac{\pi }{3} - x} \right)\\ = \dfrac{{1 - \cos 2x}}{2} + \dfrac{{1 - \cos \left( {\dfrac{{2\pi }}{3} - 2x} \right)}}{2} + \dfrac{1}{2}\left[ {\cos \left( {2x - \dfrac{\pi }{3}} \right) - \cos \dfrac{\pi }{3}} \right]\\ = \dfrac{{1 - \cos 2x + 1 - \cos \left( {\pi - \dfrac{\pi }{3} - 2x} \right) + \cos \left( {2x - \dfrac{\pi }{3}} \right) - \cos \dfrac{\pi }{3}}}{2}\\ = \dfrac{{1 - \cos 2x + 1 + \cos \left( {\dfrac{\pi }{3} + 2x} \right) + \cos \left( {\dfrac{\pi }{3} - 2x} \right) - \cos \dfrac{\pi }{3}}}{2}\\ = \dfrac{{1 - \cos 2x + 1 + 2\cos 2x\cos \dfrac{\pi }{3} - \dfrac{1}{2}}}{2} = \dfrac{{1 - \cos 2x + 1 + \cos 2x - \dfrac{1}{2}}}{2} = \dfrac{3}{4} \end{array}$
