\(\sin ^{2} x+\cos ^{2} 2x+\sin ^{2} 3x+\cos ^{2} 4x=2.\)
\(\sin ^{2} x+\cos ^{2} 2x+\sin ^{2} 3x+\cos ^{2} 4x=2\Leftrightarrow \frac{1-\cos 2x}{2} +\frac{1+\cos 4x}{2} +\frac{1-\cos 6x}{2} +\frac{1+\cos 8x}{2} =2 \)
\(\Leftrightarrow \left(\cos 8x-\cos 2x\right)-\left(\cos 6x-\cos 4x\right)=0 \)
\(\Leftrightarrow -2\sin 5x.\sin 3x+2\sin 5x.\sin x=0 \)
\(\Leftrightarrow \sin 5x\left(\sin x-\sin 3x\right)=0 \)
\(\Leftrightarrow \left[\begin{array}{l} {\sin 5x=0} \\ {\sin 3x=\sin x} \end{array}\right. \Leftrightarrow \left[\begin{array}{l} {5x=k\pi } \\ {3x=x+k2\pi } \\ {3x=\pi -x+k2\pi } \end{array}\right.\)
\(\Leftrightarrow \left[\begin{array}{l} {x=\frac{k\pi }{5} } \\ {x=k\pi } \\ {x=\frac{\pi }{4} +\frac{k\pi }{2} } \end{array}\right. \Leftrightarrow \left[\begin{array}{l} {x=\frac{k\pi }{5} } \\ {x=\frac{\pi }{4} +\frac{k\pi }{2} } \end{array}\right. \, \left(k\in {\rm Z}\right).
\)
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