1.
\(sin^2x+sin^22x+sin^23x=\dfrac{3}{2}\)
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{2}cos2x+\dfrac{1}{2}-\dfrac{1}{2}cos4x+\dfrac{1}{2}-\dfrac{1}{2}cos6x=\dfrac{3}{2}\)
\(\Leftrightarrow cos2x+cos6x+cos4x=0\)
\(\Leftrightarrow2cos4x.cos2x+cos4x=0\)
\(\Leftrightarrow cos4x\left(2cos2x+cos4x\right)=0\)
\(\Leftrightarrow cos4x\left(2cos2x+2cos^22x-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos4x=0\\2cos^22x+2cos2x-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cos4x=0\\cos2x=\dfrac{-1+\sqrt{3}}{2}\\cos2x=\dfrac{-1-\sqrt{3}}{2}< -1\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}4x=\dfrac{\pi}{2}+k\pi\\2x=\pm arcos\left(\dfrac{-1+\sqrt{3}}{2}\right)+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}\\x=\pm\dfrac{1}{2}arccos\left(\dfrac{-1+\sqrt{3}}{2}\right)+k\pi\end{matrix}\right.\)
2.
\(cos^2x+cos^22x+cos^23x=1\)
\(\Leftrightarrow\dfrac{1}{2}+\dfrac{1}{2}cos2x+cos^22x+\dfrac{1}{2}+\dfrac{1}{2}cos6x=1\)
\(\Leftrightarrow cos2x+cos6x+2cos^22x=0\)
\(\Leftrightarrow2cos2x.cos4x+2cos^22x=0\)
\(\Leftrightarrow cos2x\left(cos2x+cos4x\right)=0\)
\(\Leftrightarrow cos2x\left(cos2x+2cos^22x-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=0\\2cos^22x+cos2x-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=0\\cos2x=-1\\cos2x=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{\pi}{2}+k\pi\\2x=\pi+k2\pi\\2x=\pm\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x=\dfrac{\pi}{2}+k\pi\\x=\pm\dfrac{\pi}{6}+k\pi\end{matrix}\right.\)
