$\begin{array}{l}
\sin 4x - \cos 6x = \sqrt 3 \left( {\sin 6x + \cos 4x} \right)\\
\Leftrightarrow \dfrac{1}{2}\sin 4x - \dfrac{{\sqrt 3 }}{2}\cos 4x = \dfrac{{\sqrt 3 }}{2}\sin 6x + \dfrac{1}{2}\cos 6x\\
\Leftrightarrow \sin \left( {4x - \dfrac{\pi }{3}} \right) = \sin \left( {6x + \dfrac{\pi }{6}} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
4x - \dfrac{\pi }{3} = 6x + \dfrac{\pi }{6} + k2\pi \\
4x - \dfrac{\pi }{3} = \pi - 6x - \dfrac{\pi }{6} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
2x = - \dfrac{\pi }{2} - k2\pi \\
10x = \dfrac{{7\pi }}{6} + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = - \dfrac{\pi }{4} + k\pi \\
x = \dfrac{{7\pi }}{{60}} + \dfrac{{k\pi }}{5}
\end{array} \right.\left( {k \in \mathbb{Z}} \right)
\end{array}$
