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