1.
$\sin x+\sin3x+\sin2x=0$
$\to 2\sin2x.\cos x+\sin2x=0$
$\to \sin2x(2\cos x+1)=0$
$\to \left[ \begin{array}{l}
\sin2x=0\\
\cos x=\dfrac{-1}{2}\end{array} \right.$
$\to \left[ \begin{array}{l}x=\dfrac{k\pi}{2}\\
x=\pm\dfrac{2\pi}{3}+k2\pi\end{array} \right.$
2.
$\sin x+\sin4x+\sin2x+\sin3x=0$
$\to 2\sin\dfrac{5x}{2}\cos\dfrac{3x}{2}+2\sin\dfrac{5x}{2}\sin\dfrac{x}{2}=0$
$\to 2\sin\dfrac{5x}{2}\left( \cos\dfrac{3x}{2}+\cos\dfrac{x}{2}\right)=0$
$\to 2\sin\dfrac{5x}{2}.2\cos x.\cos\dfrac{x}{2}=0$
$\to \left[\begin{array}{l} \sin x=\dfrac{5x}{2}\\
\cos x=0\\
\cos\dfrac{x}{2}=0\end{array}\right.$
$\to \left[\begin{array}{l} x=\dfrac{k2\pi}{5}\\
x=\dfrac{\pi}{2}+k\pi\\
x=\pi+k2\pi \end{array}\right.$
3.
$\sin^2x+\sin^23x=\cos^22x+\cos^24x$
$\to \dfrac{1-\cos2x+1-\cos6x}{2}=\dfrac{1+\cos4x+1+\cos8x}{2}$
$\to \cos2x+\cos8x+\cos4x+\cos6x=0$
$\to 2\cos5x.\cos3x+2\cos5x.\cos x=0$
$\to 2\cos5x(\cos3x+\cos x)=0$
$\to 2\cos5x.2\cos2x.\cos x=0$
$\to \left[\begin{array}{l} \cos5x=0 \\
\cos2x=0 \\
\cos x=0\\
\end{array}\right.$
$\to \left[\begin{array}{I}
x=\dfrac{\pi}{10}+\dfrac{k\pi}{5}\\
x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\
x=\dfrac{\pi}{2}+k\pi
\end{array}\right.$
