`~rai~`
\(\cos\left(x+\dfrac{\pi}{3}\right)+\sin\left(\dfrac{\pi}{2}+2x\right)=0\\\Leftrightarrow \cos\left(x+\dfrac{\pi}{3}\right)+\cos2x=0\\\Leftrightarrow \cos2x=-\cos\left(x+\dfrac{\pi}{3}\right)\\\Leftrightarrow \cos2x=\cos\left(\pi-x-\dfrac{\pi}{3}\right)\\\Leftrightarrow \cos2x=\cos\left(\dfrac{2\pi}{3}-x\right)\\\Leftrightarrow \left[\begin{array}{I}2x=\dfrac{2\pi}{3}-x+k2\pi\\2x=x-\dfrac{2\pi}{3}+k2\pi\end{array}\right.\\\Leftrightarrow \left[\begin{array}{I}3x=\dfrac{2\pi}{3}+k2\pi\\x=-\dfrac{2\pi}{3}+k2\pi\end{array}\right.\\\Leftrightarrow \left[\begin{array}{I}x=\dfrac{2\pi}{9}+k\dfrac{2\pi}{3}\\x=-\dfrac{2\pi}{3}+k2\pi.\end{array}\right.\quad(k\in\mathbb{Z})\)
