\[\begin{array}{l}
{\sin ^4}x + {\cos ^4}x = {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^2} - 2{\sin ^2}x{\cos ^2}x\\
= 1 - 2{\sin ^2}x{\cos ^2}x = 1 - \frac{1}{2}.\left( {4{{\sin }^2}x{{\cos }^2}x} \right) = 1 - \frac{1}{2}{\sin ^2}2x\\
{\sin ^6}x + {\cos ^6}x = {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^3} - 3{\sin ^2}x{\cos ^2}x\left( {{{\sin }^2}x + {{\cos }^2}x} \right)\\
= 1 - 3{\sin ^2}x{\cos ^2}x = 1 - \frac{3}{4}.\left( {4{{\sin }^2}x{{\cos }^2}x} \right) = 1 - \frac{3}{4}{\sin ^2}2x
\end{array}\]
