a, \(sinx+cosx=1+sin2x\)
\(\Leftrightarrow sinx+cosx=sin^2x+cos^2x+2sinx.cosx\)
\(\Leftrightarrow sinx+cosx=\left(sinx+cosx\right)^2\)
\(\Leftrightarrow\left(sinx+cosx\right)\left(sinx+cosx-1\right)=0\)
\(\Leftrightarrow\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right).\left[\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)-1\right]=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(x+\dfrac{\pi}{4}\right)=0\\sin\left(x+\dfrac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\\x=k2\pi\\x=\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
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