5. Given that $S-A B C$ is a tetrahedron 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) $(1,2 \sqrt{2})$
5.A.

Through point $O$, construct planes parallel to $S A$, $S B$, and $S C$, forming a rectangular parallelepiped with $O S$ as its diagonal. Then, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$.
Thus, $\sin ^{2} \alpha=\cos ^{2} \beta+\cos ^{2} \gamma \geqslant 2 \cos \gamma \cdot \cos \beta$.
Similarly, $\sin ^{2} \beta=\cos ^{2} \alpha+\cos ^{2} \gamma \geqslant 2 \cos \alpha \cdot \cos \gamma$,
$$
\sin ^{2} \gamma=\cos ^{2} \beta+\cos ^{2} \alpha \geqslant 2 \cos \alpha \cdot \cos \beta \text {. }
$$

Therefore, $\tan \alpha \cdot \tan \beta \cdot \tan \gamma \geqslant 2 \sqrt{2}$.