$\begin{array}{l}
a)\cos x\left( {\sin 2x + 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\sin 2x + 1 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\sin 2x = - 1
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k\pi \\
2x = - \dfrac{\pi }{2} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k\pi \\
x = - \dfrac{\pi }{4} + k\pi
\end{array} \right.\left( {k \in Z} \right)\\
b)\left( {\sin x + \cos x} \right)\left( {2\cos 2x - 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x + \cos x = 0\\
2\cos 2x - 1 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\sqrt 2 \sin \left( {x + \dfrac{\pi }{4}} \right) = 0\\
\cos 2x = \dfrac{1}{2}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\sin \left( {x + \dfrac{\pi }{4}} \right) = 0\\
\cos 2x = \cos \dfrac{\pi }{3}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x + \dfrac{\pi }{4} = k\pi \\
2x = \pm \dfrac{\pi }{3} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = - \dfrac{\pi }{4} + k\pi \\
x = \pm \dfrac{\pi }{6} + k\pi
\end{array} \right.\left( {k \in Z} \right)
\end{array}$
