$\begin{array}{l}
\cos 3x - \sin x = \sqrt 3 \left( {\cos x - \sin 3x} \right)\\
\Leftrightarrow \cos 3x + \sqrt 3 \sin 3x = \sin x + \sqrt 3 \cos x\\
\Leftrightarrow \frac{{\sqrt 3 }}{2}\sin 3x + \frac{1}{2}\cos 3x = \frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x\\
\Leftrightarrow \sin \left( {3x + \frac{\pi }{6}} \right) = \sin \left( {x + \frac{\pi }{3}} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
3x + \frac{\pi }{6} = x + \frac{\pi }{3} + k2\pi \\
3x + \frac{\pi }{6} = \pi - x - \frac{\pi }{3} + m2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{{12}} + k\pi \\
x = \frac{\pi }{8} + \frac{{m\pi }}{2}
\end{array} \right.\,\,\,\left( {k,\,\,m \in Z} \right)\\
b = \frac{\pi }{6};\,\,d = \frac{\pi }{3} \Rightarrow b + d = \frac{\pi }{2}.
\end{array}$
