a. cos2x + cos4x + cos6x = 0
\(\Leftrightarrow\left(cos2x+cos6x\right)+cos4x=0\\ \Leftrightarrow2cos4x.cos2x+cos4x=0\\ \Leftrightarrow cos4x\left(2cos2x+1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}cos4x=0\\cos2x=\dfrac{-1}{2}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}\\x=\pm\dfrac{\pi}{3}+k\pi\end{matrix}\right.\left(k\in Z\right)}\)
1.
\(cos2x+cos6x+cos4x=0\)
\(\Leftrightarrow2cos4x.cos2x+cos4x=0\)
\(\Leftrightarrow cos4x\left(2cos2x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos4x=0\\cos2x=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}4x=\dfrac{\pi}{2}+k\pi\\2x=\pm\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}\\x=\pm\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
2.
\(\Leftrightarrow1+cos2x+cosx+cos3x=0\)
\(\Leftrightarrow1+2cos^2x-1+2cos2x.cosx=0\)
\(\Leftrightarrow cos^2x+cos2x.cosx=0\)
\(\Leftrightarrow cosx\left(cos2x+cosx\right)=0\)
\(\Leftrightarrow cosx\left(2cos^2x+cosx-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\cosx=-1\\cosx=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\pi+k2\pi\\x=\pm\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
3.
\(\Leftrightarrow cos3x-sin3x=cosx-sinx\)
\(\Leftrightarrow\sqrt{2}cos\left(3x+\dfrac{\pi}{4}\right)=\sqrt{2}cos\left(x+\dfrac{\pi}{4}\right)\)
\(\Leftrightarrow cos\left(3x+\dfrac{\pi}{4}\right)=cos\left(x+\dfrac{\pi}{4}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}3x+\dfrac{\pi}{4}=x+\dfrac{\pi}{4}+k2\pi\\3x+\dfrac{\pi}{4}=-x-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=k2\pi\\4x=-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=-\dfrac{\pi}{8}+\dfrac{k\pi}{2}\end{matrix}\right.\)
