$\begin{array}{l}
{\cos ^2}x + \sin x + 1 = 0\\
\Leftrightarrow 1 - {\sin ^2}x + \sin x + 1 = 0\\
\Leftrightarrow {\sin ^2}x - \sin x - 2 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = 2\,\,\left( {ktm} \right)\\
\sin x = - 1
\end{array} \right. \Leftrightarrow x = - \frac{\pi }{2} + k2\pi \,\,\left( {k \in Z} \right).
\end{array}$