6.
\(cos^2x+cos^22x+cos^23x+cos^24x=2\)
\(\Leftrightarrow2cos^2x-1+2cos^22x-1+2cos^23x-1+2cos^24x-1=0\)
\(\Leftrightarrow cos2x+cos4x+cos6x+cos8x=0\)
\(\Leftrightarrow2cos5x.cos3x+2cos5x.cosx=0\)
\(\Leftrightarrow cos5x.\left(cos3x+cosx\right)=0\)
\(\Leftrightarrow2cos5x.cos2x.cosx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos5x=0\\cos2x=0\\cosx=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}5x=\dfrac{\pi}{2}+k\pi\\2x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{10}+\dfrac{k\pi}{5}\\x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
