1.
\(sinx+sin5x+sin3x=0\)
\(\Leftrightarrow2sin3x.cos2x+sin3x=0\)
\(\Leftrightarrow sin3x\left(2cos2x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin3x=0\\cos2x=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}3x=k\pi\\2x=\pm\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{3}\\x=\pm\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
2.
\(cos7x-cos3x+sin8x+sin2x=0\)
\(\Leftrightarrow-2sin5x.sin2x+2sin5x.cos3x=0\)
\(\Leftrightarrow sin5x\left(sin2x-cos3x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin5x=0\\cos3x=sin2x=cos\left(\dfrac{\pi}{2}-2x\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}5x=k\pi\\3x=\dfrac{\pi}{2}-2x+k2\pi\\3x=2x-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{5}\\x=\dfrac{\pi}{10}+\dfrac{k2\pi}{5}\\x=-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
