This problem can be solved by using the telescoping technique. \begin{align} &\cos\frac{2\pi}{7} \cdot \cos \frac{4\pi}{7}\cdot \cos \frac{8\pi}{7} \\ &=\frac{1}{\sin\frac{2\pi}{7}}\cdot\sin\frac{2\pi}{7}\cdot\cos\frac{2\pi}{7}\cdot \cos\frac{4\pi}{7}\cdot \cos \frac{8\pi}{7}\\ &=\frac{1}{\sin\frac{2\pi}{7}}\cdot\frac{1}{2}\cdot\sin\frac{4\pi}{7}\cdot \cos\frac{4\pi}{7}\cdot \cos \frac{8\pi}{7}\\ &=\frac{1}{\sin\frac{2\pi}{7}}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\sin\frac{8\pi}{7}\cdot \cos \frac{8\pi}{7}\\ &=\frac{1}{\sin\frac{2\pi}{7}}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\sin\frac{16\pi}{7}\\ &=\frac{1}{\sin\frac{2\pi}{7}}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\sin\frac{2\pi}{7}\\ &=\boxed{\frac{1}{8}} \end{align}