Phân tích sin^8 x + cos^8 x

Đề bài: Phân tích ${\sin }^{8}x+co{s}^{8}x$.

Trả lời

Hướng dẫn giải:

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

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