\(sinx-\sqrt{3}cos\left(x+\pi\right)=2sin2x\)
\(\Leftrightarrow sinx+\sqrt{3}cosx=2sin2x\)
\(\Leftrightarrow\dfrac{1}{2}sinx+\dfrac{\sqrt{3}}{2}cosx=sin2x\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{3}\right)=sin2x\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=x+\dfrac{\pi}{3}+k2\pi\\2x=\dfrac{2\pi}{3}-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{3}+k2\pi\\x=\dfrac{2\pi}{9}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
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