6. If $\tan x_{1} \cdot \tan x_{2} \cdots \cdots \tan x_{2012}=1$, then the maximum value of $\sin x_{1} \cdot \sin x_{2} \cdots \cdots \sin x_{2012}$ is
6. $\frac{1}{2^{1006}}$.

From $\tan x_{1} \cdot \tan x_{2} \cdots \cdot \tan x_{2012}=1$, we get
$$
\begin{array}{l}
\sin x_{1} \cdot \sin x_{2} \cdots \cdots \sin x_{2012} \\
=\cos x_{1} \cdot \cos x_{2} \cdots \cdots \cos x_{2012} .
\end{array}
$$

Therefore, $2^{2012}\left(\sin x_{1} \cdot \sin x_{2} \cdots \cdot \sin x_{2012}\right)^{2}$
$$
=\sin 2 x_{1} \cdot \sin 2 x_{2} \cdots \cdot \sin 2 x_{2012} \leqslant 1 \text {. }
$$

Thus, $\sin x_{1} \cdot \sin x_{2} \cdots \cdot \sin x_{2012} \leqslant \frac{1}{2^{1006}}$, when $x_{1}=x_{2}=\cdots=x_{2012}=\frac{\pi}{4}$, the equality holds. Therefore, the maximum value sought is $\frac{1}{2^{1006}}$.