5. Let $a_{1}=$
$\frac{\pi}{6}, a_{n} \in\left(0, \frac{\pi}{2}\right)$ and $\tan a_{n+1} \cdot \cos a_{n}=1, n \geqslant 1$. If $\prod_{k=1}^{m} \sin a_{k}=\frac{1}{100}$, then $m=$ $\qquad$
$$
\begin{array}{l}
\tan a_{n+1} \cdot \cos a_{n}=1 \Rightarrow \tan ^{2} a_{n+1}=\frac{1}{\cos ^{2} a_{n}}=\tan ^{2} a_{n}+1 \\
\Rightarrow \tan ^{2} a_{n}=\tan ^{2} a_{1}+n-1=\frac{3 n-2}{3} \Rightarrow \sin a_{n}=\sqrt{\frac{3 n-2}{3 n+1}}, \text{ then } \\
\prod_{k=1}^{m} \sin a_{k}=\prod_{k=1}^{m} \sqrt{\frac{3 k-2}{3 k+1}}=\sqrt{\frac{1}{3 m+1}}=\frac{1}{100} \Rightarrow 3 m+1=10000 \Rightarrow m=3333 .
\end{array}
$$