\(32sin^6\dfrac{x}{2}+sin3x=3sinx\)
\(\Leftrightarrow32sin^6\dfrac{x}{2}+3sinx-4sin^3x=3sinx\)
\(\Leftrightarrow8sin^6\dfrac{x}{2}=sin^3x\)
\(\Leftrightarrow8sin^6\dfrac{x}{2}=8sin^3\dfrac{x}{2}.cos^3\dfrac{x}{2}\)
\(\Leftrightarrow sin^3\dfrac{x}{2}\left(1-cos^3\dfrac{x}{2}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\dfrac{x}{2}=0\\cos\dfrac{x}{2}=1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}=k\pi\\\dfrac{x}{2}=k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=k4\pi\end{matrix}\right.\)
\(\Leftrightarrow x=k2\pi\)
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