\(\Leftrightarrow cos^2x-sin^2x+cos^3x-sin^3x+cos^4x-sin^4x=0\)
\(\Leftrightarrow cos^2x-sin^2x+\left(cosx-sinx\right)\left(1+sinx.cosx\right)+\left(cos^2x-sin^2x\right)\left(cos^2x+sin^2x\right)=0\)
\(\Leftrightarrow\left(cosx-sinx\right)\left(2sinx+2cosx\right)+\left(cosx-sinx\right)\left(1+sinx.cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx-sinx=0\Rightarrow x=\dfrac{\pi}{4}+k\pi\\2sinx+2cosx+1+sinx.cosx=0\left(1\right)\end{matrix}\right.\)
Xét (1) , đặt \(sinx+cosx=t\Rightarrow\left|t\right|\le\sqrt{2}\)
\(sinx.cosx=\dfrac{t^2-1}{2}\)
(1) \(\Leftrightarrow2t+1+\dfrac{t^2-1}{2}=0\)
\(\Leftrightarrow t^2+4t+1=0\Rightarrow\left[{}\begin{matrix}t=-2-\sqrt{3}\left(loại\right)\\t=-2+\sqrt{3}\end{matrix}\right.\)
\(\Rightarrow\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)=-2+\sqrt{3}\)
\(\Rightarrow sin\left(x+\dfrac{\pi}{4}\right)=\dfrac{\sqrt{6}-2\sqrt{2}}{2}\)
\(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+arcsin\left(\dfrac{\sqrt{6}-2\sqrt{2}}{2}\right)+k2\pi\\x=\dfrac{3\pi}{4}-arcsin\left(\dfrac{\sqrt{6}-2\sqrt{2}}{2}\right)+k2\pi\end{matrix}\right.\)
