1.
\(pt\Leftrightarrow sin4x\left(sin5x+sin3x\right)=sin2x.sinx\)
\(\Leftrightarrow2sin^24x.cosx=sin2x.sinx\)
\(\Leftrightarrow2sin^24x.cosx=2sin^2x.cosx\)
\(\Leftrightarrow2cosx.\left(sin^24x-sin^2x\right)=0\)
\(\Leftrightarrow2cosx.\left(sin4x-sinx\right)\left(sin4x+sinx\right)=0\)
\(\Leftrightarrow8cosx.sin\dfrac{5x}{2}.cos\dfrac{3x}{2}.sin\dfrac{5x}{2}.cos\dfrac{3x}{2}=0\)
\(\Leftrightarrow8cosx.sin5x.sin3x=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sin5x=0\\sin3x=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{k\pi}{5}\\x=\dfrac{k\pi}{3}\end{matrix}\right.\)
\(pt\Leftrightarrow sin8x+sin2x=sin16x+sin2x\)
\(\Leftrightarrow sin8x=2sin8x.cos8x\)
\(\Leftrightarrow sin8x\left(1-2cos8x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin8x=0\\cos8x=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}8x=k\pi\\8x=\pm\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{8}\\x=\pm\dfrac{\pi}{24}+\dfrac{k\pi}{4}\end{matrix}\right.\)
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