1. Given the condition $p: \sqrt{1+\sin 2 \alpha}=\frac{4}{3}$ and the condition $q:|\sin \alpha+\cos \alpha|=\frac{4}{3}$. Then $p$ is ( ) of $q$.
(A) a sufficient but not necessary condition
(B) a necessary but not sufficient condition
(C) a necessary and sufficient condition
(D) neither a sufficient nor a necessary condition
$-1 . \mathrm{C}$.
$$
\begin{array}{l}
\text { Since } \sqrt{1+\sin 2 \alpha}=\sqrt{(\sin \alpha+\cos \alpha)^{2}} \\
=|\sin \alpha+\cos \alpha|,
\end{array}
$$

Therefore, $p$ is a necessary and sufficient condition for $q$.