a, \(\Leftrightarrow\left(sinx+sin2x\right)+\left(cos2x+cosx+1\right)=0\Leftrightarrow\left(sinx+2.sinx.cosx\right)+\left(2cos^2x-1+cosx+1\right)=0\Leftrightarrow sinx\left(2cosx+1\right)+cosx\left(2cosx+1\right)=0\Leftrightarrow\left(2cosx+1\right)\left(sinx+cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}2cosx=-1\\sinx+cosx=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}cosx=-\dfrac{1}{2}\\\sqrt{2}.sin\left(x+\dfrac{\pi}{4}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{3\pi}{4}+k2\pi\\x=-\dfrac{\pi}{4}+k\pi\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{3\pi}{4}+k2\pi\\x=-\dfrac{\pi}{4}+k\pi\end{matrix}\right.\) (k∈Z)
b, \(\Leftrightarrow\left(cos3x-cos2x\right)+\left(cos2x-1\right)=0\Leftrightarrow-2sinx.sin2x-2sin^2x=0\Leftrightarrow-2sinx.\left(sin2x+sinx\right)=0\Leftrightarrow-2sinx.2sin\dfrac{3x}{2}.sin\dfrac{x}{2}=0\) \(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\sin\dfrac{x}{2}=0\\sin\dfrac{3x}{2}=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=2k\pi\\x=\dfrac{2k\pi}{3}\end{matrix}\right.\) \(\Leftrightarrow x=\dfrac{2k\pi}{3},k\in Z\)
