1 + cos4x = cos2x \( \Leftrightarrow 1 + 2{\cos ^2}2x - 1 = \cos 2x\)

\( \Leftrightarrow 2{\cos ^2}2x - \cos 2x = 0 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{\cos 2x = 0}\\{\cos 2x = \frac{1}{2}}\end{array}} \right.\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{2x = \frac{\pi }{2} + k\pi }\\{2x = \pm \frac{\pi }{3} + k2\pi }\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k\frac{\pi }{2}}\\{x = \pm \frac{\pi }{6} + k\pi }\end{array}} \right.\left( {k \in \mathbb{Z}} \right)\).