Phân tích \({\sin ^8}x + co{s^8}x\).

\(\begin{array}{l}{\sin ^8}x + co{s^8}x\\ = {\left( {{{\sin }^4}x + co{s^4}x} \right)^2} - 2{\sin ^4}x.co{s^4}x\\ = {\left( {1 - 2{{\sin }^2}x.co{s^2}x} \right)^2} - \frac{1}{8}{\sin ^4}2x\\ = {\left( {1 - \frac{1}{2}{{\sin }^2}2x} \right)^2} - \frac{1}{8}{\sin ^4}2x\\ = 1 - {\sin ^2}2x + \frac{1}{8}{\sin ^4}2x\end{array}\)

\(\begin{array}{l} = 1 - \frac{{1 - cos4x}}{2} + \frac{1}{8}{\left( {\frac{{1 - cos4x}}{2}} \right)^2}\\ = 1 - \frac{{1 - cos4x}}{2} + \frac{1}{{32}}\left( {1 - 2cos4x + \frac{{1 + \cos 8x}}{2}} \right)\\ = \frac{{35}}{{64}} + \frac{7}{{16}}cos4x + \frac{1}{{64}}cos8x\end{array}\)