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