Hướng dẫn giải:

$\begin{align}& 2{{\cos }^{3}}x+\cos 2x+\sin x=0 \\& \Leftrightarrow 2{{\cos }^{2}}x(\cos x+1)-(1-\sin x)=0 \\& \Leftrightarrow \left[ \begin{matrix}1-\sin x=0  \\2\sin x+2\cos x+2\sin x\cos x+1=0  \\\end{matrix} \right. \\\end{align}$

$+)1-\sin x=0\Rightarrow x=\frac{\pi }{2}+k2\pi ,k\in \mathbb{Z}$.

$+)\text{ }2\sin x+2\cos x+2\sin x\cos x+1=0$

Đặt $t=\sin x+\cos x=\sqrt{2}\sin \left( x+\frac{\pi }{4} \right)\left( \left| t \right|\le \sqrt{2} \right)$ ta có: $\sin x\cos x=\frac{{{t}^{2}}-1}{2}$.

$\begin{align}& \Rightarrow 2t+{{t}^{2}}-1+1=0 \\& \Leftrightarrow \left[ \begin{matrix}t=0  \\t=-2(l)  \\\end{matrix} \right. \\& \Rightarrow \sin (x+\frac{\pi }{4})=0 \\& \Leftrightarrow x=\frac{-\pi }{4}+k\pi ,k\in \mathbb{Z} \\\end{align}$