$\begin{array}{l}
{\cos ^6}x + {\sin ^6}x = 2\left( {{{\cos }^8}x + {{\sin }^8}x} \right)\\
\Leftrightarrow {\cos ^6}x + {\sin ^6}x = 2{\cos ^8}x + 2{\sin ^8}x\\
\Leftrightarrow {\cos ^6}x - 2{\cos ^8}x + {\sin ^6}x - 2{\sin ^8}x = 0\\
\Leftrightarrow {\cos ^6}x\left( {1 - 2{{\cos }^2}x} \right) + {\sin ^6}x\left( {1 - 2{{\sin }^2}x} \right) = 0\\
\Leftrightarrow - \cos 2x.{\cos ^6}x + {\sin ^6}x.\cos 2x = 0\\
\Leftrightarrow \cos 2x\left( {{{\sin }^6}x - {{\cos }^6}x} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
{\sin ^6}x - {\cos ^6}x = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
{\cos ^6}x - {\sin ^6}x = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
\left( {{{\cos }^2}x - {{\sin }^2}x} \right)\left( {{{\cos }^4}x + {{\sin }^4}x + {{\sin }^2}x{{\cos }^2}x} \right) = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
\cos 2x\left[ {{{\left( {{{\sin }^2}x + {{\cos }^2}x} \right)}^2} - {{\sin }^2}x{{\cos }^2}x} \right] = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
1 - {\sin ^2}x{\cos ^2}x = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
1 - \dfrac{1}{4}{\sin ^2}2x = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
\dfrac{1}{4}{\sin ^2}2x = 1
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
{\sin ^2}2x = 4
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 0\\
\sin 2x = - 2(L)\\
\sin 2x = 2(L)
\end{array} \right. \Rightarrow \cos 2x = 0\\
\Rightarrow 2x = \dfrac{\pi }{2} + k\pi \Leftrightarrow x = \dfrac{\pi }{4} + \dfrac{{k\pi }}{2}\left( {k \in \mathbb{Z}} \right)
\end{array}$
