1.
\(cos^23x+cos^24x+cos^25x=\dfrac{3}{2}\)
\(\Leftrightarrow2cos^23x-1+2cos^24x-1+2cos^25x-1=0\)
\(\Leftrightarrow cos6x+cos8x+cos10x=0\)
\(\Leftrightarrow2cos8x.cos2x+cos8x=0\)
\(\Leftrightarrow cos8x\left(2cos2x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos8x=0\\cos2x=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{16}+\dfrac{k\pi}{8}\\x=\pm\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
2.
\(3cos^22x-2sin^2x+cos^2x=0\)
\(\Leftrightarrow6cos^22x-4sin^2x+2cos^2x=0\)
\(\Leftrightarrow6cos^22x+1-2sin^2x+2\left(cos^2x-sin^2x\right)=1\)
\(\Leftrightarrow6cos^22x+cos2x+4cos2x=1\)
\(\Leftrightarrow6cos^22x+5cos2x-1=0\)
\(\Leftrightarrow\left(cos2x+1\right)\left(6cos2x-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=-1\\cos2x=\dfrac{1}{6}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=\pi+k2\pi\\2x=\pm arccos\left(\dfrac{1}{6}\right)+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}\pi+k\pi\\x=\pm\dfrac{1}{2}arccos\left(\dfrac{1}{6}\right)+k\pi\end{matrix}\right.\)
