\(4cosx-2cos2x-cos4x=1\)
\(\Leftrightarrow4cosx-2cos2x-\left(2cos^22x-1\right)=1\)
\(\Leftrightarrow4cosx-2cos2x-2cos^22x=0\)
\(\Leftrightarrow4cosx-2cos2x\cdot\left(1+cos2x\right)=0\)
\(\Leftrightarrow4cosx-2cos2x\cdot2cos^2x=0\)
\(\Leftrightarrow2cosx\cdot\left(2-2cos2x\cdot cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\rightarrow x=\dfrac{\pi}{2}+k\pi\left(k\in Z\right)\\2-2cos2x\cdot cosx=0\end{matrix}\right.\)
\(\Leftrightarrow2cos2x\cdot cosx=2\)
\(\Leftrightarrow cos2x\cdot cosx=1\)
\(\Leftrightarrow\left(2cos^2x-1\right)\cdot cosx-1=0\)
\(\Leftrightarrow2cos^3x-cosx-1=0\)
\(\Leftrightarrow cosx=1\)
\(\Leftrightarrow x=k2\pi\) \(\left(k\in Z\right)\)
