\(sin\left(2x+\dfrac{\pi}{4}\right)+cosx=0\)
\(\Leftrightarrow cos\left(2x-\dfrac{\pi}{4}\right)+cosx=0\)
\(\Leftrightarrow2cos\left(\dfrac{3x}{2}-\dfrac{\pi}{8}\right).cos\left(\dfrac{x}{2}-\dfrac{\pi}{8}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\left(\dfrac{3x}{2}-\dfrac{\pi}{8}\right)=0\\cos\left(\dfrac{x}{2}-\dfrac{\pi}{8}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{3x}{2}-\dfrac{\pi}{8}=\dfrac{\pi}{2}+k\pi\\\dfrac{x}{2}-\dfrac{\pi}{8}=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+\dfrac{k2\pi}{3}\\x=\dfrac{5\pi}{4}+k2\pi\end{matrix}\right.\)
