Example 8
$$
\begin{aligned}
\frac{\cos \left(30^{\circ}-n^{\circ}\right)}{\cos n^{\circ}} & =\frac{\cos 30^{\circ} \cos n^{\circ}+\sin 30^{\circ} \sin n^{\circ}}{\cos n^{\circ}}=\cos 30^{\circ}+\sin 30^{\circ} \tan n^{\circ} \\
& =\frac{1}{2}\left(\sqrt{3}+\tan n^{\circ}\right) .
\end{aligned}
$$
If we take $n=1,2,3, \cdots, 29$, then $30-n$ also takes all values $1,2,3, \cdots, 29$. Thus, we have: $\left(\sqrt{3}+\tan 1^{\circ}\right)\left(\sqrt{3}+\tan 2^{\circ}\right) \cdots\left(\sqrt{3}+\tan 29^{\circ}\right)=2^{29}$.