$\begin{array}{l}
\tan \dfrac{{5\pi }}{{12}} = \tan \left( {\dfrac{{2\pi }}{3} - \dfrac{\pi }{4}} \right)\\
= \dfrac{{\tan \dfrac{{2\pi }}{3} - \tan \dfrac{\pi }{4}}}{{1 + \tan \dfrac{{2\pi }}{3}\tan \dfrac{\pi }{4}}} = \dfrac{{ - \sqrt 3 - 1}}{{1 - \sqrt 3 }} = \dfrac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}\\
\left( {\sqrt 3 - 1} \right)\sin x - \left( {\sqrt 3 + 1} \right)\cos x + \sqrt 3 - 1 = 0\\
\Leftrightarrow \sin x - \dfrac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}\cos x + 1 = 0\\
\Leftrightarrow \sin x - \tan \dfrac{{5\pi }}{{12}}\cos x + 1 = 0\\
\Leftrightarrow \cos \dfrac{{5\pi }}{{12}}\sin x - \sin \dfrac{{5\pi }}{{12}}\cos x + \cos \dfrac{{5\pi }}{{12}} = 0\\
\Leftrightarrow \sin \left( {x - \dfrac{{5\pi }}{{12}}} \right) = - \cos \dfrac{{5\pi }}{{12}}\\
\Leftrightarrow \sin \left( {x - \dfrac{{5\pi }}{{12}}} \right) = - \sin \left( {\dfrac{\pi }{{12}}} \right)\\
\Leftrightarrow \sin \left( {x - \dfrac{{5\pi }}{{12}}} \right) = \sin \left( { - \dfrac{\pi }{{12}}} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
x - \dfrac{{5\pi }}{{12}} = - \dfrac{\pi }{{12}} + k2\pi \\
x - \dfrac{{5\pi }}{{12}} = \dfrac{{13\pi }}{{12}} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{3} + k2\pi \\
x = \dfrac{{3\pi }}{2} + k2\pi
\end{array} \right.\left( {k \in \mathbb{Z}} \right)
\end{array}$
