Giải phương trình: sin^6x+cos^6x=2(sin^8x+cos^8x)

Hướng dẫn giải:

Ta có: (*) \( \Leftrightarrow {{\sin }^{6}}x-2{{\sin }^{8}}x+co{{s}^{6}}x-2co{{s}^{8}}x=0 \)

 \( \Leftrightarrow {{\sin }^{6}}x(1-2si{{n}^{2}}x)-{{\cos }^{6}}x(2{{\cos }^{2}}x-1)=0\Leftrightarrow {{\sin }^{6}}x\cos 2x-{{\cos }^{6}}x.\cos 2x=0 \)

 \( \Leftrightarrow \cos 2x({{\sin }^{6}}x-co{{s}^{6}}x)=0\Leftrightarrow \left[ \begin{align}  & \cos 2x=0 \\  & {{\sin }^{6}}x={{\cos }^{6}}x \\ \end{align} \right. \) \( \Leftrightarrow \left[ \begin{align} & \cos 2x=0 \\  & {{\tan }^{6}}x=1 \\ \end{align} \right. \)

 \( \Leftrightarrow \left[ \begin{align}  & 2x=\frac{\pi }{2}+k\pi  \\  & \tan x=\pm 1 \\ \end{align} \right. \) \( \Leftrightarrow \left[ \begin{align} & x=\frac{\pi }{4}+\frac{k\pi }{2} \\  & x=\pm \frac{\pi }{4}+k\pi  \\ \end{align} \right.\Leftrightarrow x=\frac{\pi }{4}+\frac{k\pi }{2},\text{ }k\in \mathbb{Z} \).