2. If a space diagonal of a rectangular prism forms angles $\alpha, \beta, \gamma$ with the three edges starting from the same vertex, then
$$
\frac{\cos \beta \cdot \cos \gamma}{\cos \alpha}+\frac{\cos \gamma \cdot \cos \alpha}{\cos \beta}+\frac{\cos \alpha \cdot \cos \beta}{\cos \gamma}
$$

the minimum value is $\qquad$
2. $\sqrt{3}$.

Let $y=\sum \frac{\cos \beta \cdot \cos \gamma}{\cos \alpha}$, where “$\sum$” denotes the cyclic sum.
Since $00$.
Also, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$, then
$$
\begin{aligned}
y^{2} & =\sum \frac{\cos ^{2} \beta \cdot \cos ^{2} \gamma}{\cos ^{2} \alpha}+2 \sum \cos ^{2} \alpha \\
& =\sum \frac{\cos ^{2} \beta \cdot \cos ^{2} \gamma}{\cos ^{2} \alpha}+2 \\
& =\sum \frac{\cos ^{2} \alpha}{2}\left(\frac{\cos ^{2} \beta}{\cos ^{2} \gamma}+\frac{\cos ^{2} \gamma}{\cos ^{2} \beta}\right)+2 \\
& \geqslant \sum \frac{\cos ^{2} \alpha}{2} \times 2+2=3 .
\end{aligned}
$$

Thus, $y \geqslant \sqrt{3}$, and the equality holds if and only if $\cos \alpha=\cos \beta=\cos \gamma$, at which point $y$ attains its minimum value.