5. If $\tan x_{1} \cdot \tan x_{2} \cdots \cdots \tan x_{2000}=$ 1 , then the maximum value of $\sin x_{1} \cdot \sin x_{2} \cdots \cdot \sin x_{2 \text { obo }}$ is $\qquad$ .
5. $\frac{1}{2^{1000}}$.

From $\tan x_{1} \cdot \tan x_{2} \cdots \cdot \tan x_{2000}=1$, we get
$$
\begin{array}{l}
\sin x_{1} \cdot \sin x_{2} \cdots \cdot \sin x_{2000} \\
=\cos x_{1} \cdot \cos x_{2} \cdots \cdot \cos x_{2000} .
\end{array}
$$

Thus $2^{2000}\left(\sin x_{1} \cdot \sin x_{2} \cdots \cdots \cdot \sin x_{2000}\right)^{2}$
$$
\begin{array}{l}
\quad=\sin 2 x_{1} \cdot \sin 2 x_{2} \cdots \cdots \cdot \sin 2 x_{2000} \leqslant 1 . \\
\therefore \sin x_{1} \cdot \sin x_{2} \cdots \cdot \sin x_{2000} \leqslant \frac{1}{2^{1000}} .
\end{array}
$$

Equality can hold, for example, when $x_{1}=x_{2}=\cdots=x_{2000}=\frac{\pi}{4}$.
Therefore, the maximum value of $\sin x_{1} \cdot \sin x_{2} \cdots \cdot \sin x_{2000}$ is
$$
\frac{1}{2^{1000}}
$$