6. Calculate $\cos \frac{2 \pi}{7} \cdot \cos \frac{4 \pi}{7} \cdot \cos \frac{6 \pi}{7}$

The value is
6. $\frac{1}{8}$.

Let $S=\cos \frac{2 \pi}{7} \cdot \cos \frac{4 \pi}{7} \cdot \cos \frac{6 \pi}{7}$. Then
$$
\begin{aligned}
S & =-\cos \frac{\pi}{7} \cdot \cos \frac{2 \pi}{7} \cdot \cos \frac{4 \pi}{7} \\
\Rightarrow & -S 8 \sin \frac{\pi}{7} \\
& =8 \sin \frac{\pi}{7} \cdot \cos \frac{\pi}{7} \cdot \cos \frac{2 \pi}{7} \cdot \cos \frac{4 \pi}{7} \\
& =4 \sin \frac{2 \pi}{7} \cdot \cos \frac{2 \pi}{7} \cdot \cos \frac{4 \pi}{7} \\
& =2 \sin \frac{4 \pi}{7} \cdot \cos \frac{4 \pi}{7}=\sin \frac{8 \pi}{7} \\
& =-\sin \frac{\pi}{7} \\
\Rightarrow & S=\frac{1}{8} .
\end{aligned}
$$