2. As shown in Figure $1, S-A B C$ is a triangular pyramid with three edges mutually perpendicular, and $O$ is a point within the base $A B C$. If $\angle O S A=\alpha, \angle O S B$ $=\beta, \angle O S C=\gamma$, then the range of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$ is ( ).
(A) $[2 \sqrt{2},+\infty)$
(B) $(0,2 \sqrt{2})$
(C) $[1,2 \sqrt{2}]$
(D) $(1,2 \sqrt{2})$
2.A.

As shown in Figure 6, through $O$, planes parallel to $S A$, $S B$, and $S C$ intersect $S A$, $S B$, and $S C$ at $D$, $E$, and $F$, respectively, thus forming a rectangular parallelepiped $M F S E-O N D P$ with $S O$ as the diagonal. According to the problem,
$$
\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \text {. }
$$

Furthermore, from $\sin ^{2} \gamma=\cos ^{2} \alpha$
$$
\begin{array}{l}
+\cos ^{2} \beta \geqslant 2 \cos \alpha \cdot \cos \beta, \\
\sin ^{2} \alpha=\cos ^{2} \beta+\cos ^{2} \gamma \geqslant 2 \cos \beta \cdot \cos \gamma, \\
\sin ^{2} \beta=\cos ^{2} \alpha+\cos ^{2} \gamma \geqslant 2 \cos \alpha \cdot \cos \gamma,
\end{array}
$$

we get $\tan \alpha \cdot \tan \beta \cdot \tan \gamma \geqslant 2 \sqrt{2}$.