\(\sqrt{2}sinx+2cos^2x-1-sin2x=0\)
\(\Leftrightarrow\sqrt{2}sinx+cos2x-sin2x=0\)
\(\Leftrightarrow\sqrt{2}sinx=sin2x-cos2x\)
\(\Leftrightarrow\sqrt{2}sinx=\sqrt{2}sin\left(2x-\dfrac{\pi}{4}\right)\)
\(\Leftrightarrow sinx=sin\left(2x-\dfrac{\pi}{4}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{4}=x+k2\pi\\2x-\dfrac{\pi}{4}=\pi-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k2\pi\\x=\dfrac{5\pi}{12}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
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