设 $\alpha \in\left(0, \frac{\pi}{2}\right), \beta \in\left(0, \frac{\pi}{2}\right)$, 且 $\tan \alpha=\frac{1+\sin \beta}{\cos \beta}$, 则（ ）
$\text{A.}$ $3 \alpha-\beta=\frac{\pi}{2}$
$\text{B.}$ $3 \alpha+\beta=\frac{\pi}{2}$
$\text{C.}$ $2 \alpha-\beta=\frac{\pi}{2}$
$\text{D.}$ $2 \alpha+\beta=\frac{\pi}{2}$