\[\begin{array}{l} {\cos ^6}x + {\sin ^6}x = \frac{1}{4}{\sin ^2}2x \Leftrightarrow {\left( {{{\cos }^2}x} \right)^3} + {\left( {{{\sin }^2}x} \right)^3} = {\sin ^2}x{\cos ^2}x \hfill \\ {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) \hfill \\ \left( {{{\sin }^2}x + {{\cos }^2}x} \right)\left( {{{\sin }^4}x - {{\sin }^2}x{{\cos }^2}x + {{\cos }^4}x} \right) = {\sin ^2}x{\cos ^2}x \Leftrightarrow \hfill \\ {\sin ^4}x - {\sin ^2}x{\cos ^2}x + {\cos ^4}x = {\sin ^2}x{\cos ^2}x \Leftrightarrow \hfill \\ {\sin ^4}x - 2{\sin ^2}x{\cos ^2}x + {\cos ^4}x = 0 \Leftrightarrow \hfill \\ {\left( {{{\sin }^2}x - {{\cos }^2}x} \right)^2} = 0 \Leftrightarrow {\sin ^2}x - {\cos ^2}x = 0 \Leftrightarrow \cos 2x = 0 \Leftrightarrow \hfill \\ 2x = \frac{\pi }{2} + \pi k \Leftrightarrow x = \frac{\pi }{4} + \frac{{\pi k}}{2}. \hfill \\ \end{array}\]

\[x = \frac{\pi }{4} + \frac{{\pi k}}{2}\]