$cosx + sinx = cos2x$
$\Leftrightarrow (cosx+sinx)(cosx-sinx-1)=0$
$\Leftrightarrow\left[\begin{array}{I}
\cos x+\sin x=0\\ \cos x-\sin x =1\end{array}\right.\Leftrightarrow\left[\begin{array}{I}
\tan x=-1\\ \cos(x+\frac{\pi}{4})=\frac{1}{\sqrt{2}}\end{array}\right.$
$\Leftrightarrow\left[\begin{array}{I}
x=\frac{-\pi}{4}+k\pi\\ x=k2\pi\\ x=\frac{-\pi}{2}+k2\pi\end{array}\right., k\in Z$