$\begin{array}{l} {\sin ^3}x - \sqrt 3 {\cos ^3}x = \sin x{\cos ^2}x - \sqrt 3 {\sin ^2}x\cos x\\ \Leftrightarrow {\sin ^3}x - \sin x{\cos ^2}x = \sqrt 3 {\cos ^3}x - \sqrt 3 {\sin ^2}x\cos x\\ \Leftrightarrow \sin x\left( {{{\sin }^2}x - {{\cos }^2}x} \right) = \sqrt 3 \cos x\left( {{{\cos }^2}x - {{\sin }^2}x} \right)\\ \Leftrightarrow \left( {{{\cos }^2}x - {{\sin }^2}x} \right)\left( {\sqrt 3 \cos x + \sin x} \right) = 0\\ \Leftrightarrow 2\cos 2x\sin \left( {x + \dfrac{\pi }{3}} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} \cos 2x = 0\\ \sin \left( {x + \dfrac{\pi }{3}} \right) = 0 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} 2x = \dfrac{\pi }{2} + k\pi \\ x + \dfrac{\pi }{3} = k\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{\pi }{4} + \dfrac{{k\pi }}{2}\\ x = - \dfrac{\pi }{3} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \end{array}\\\to D$
