\(\cos\frac{2\pi}{7}.\cos\frac{4\pi}{7}.\cos\frac{6\pi}{7}\\ =-\cos\frac{2\pi}{7}.\cos\frac{4\pi}{7}.\cos\left(\pi-\frac{6\pi}{7} \right)\\ =-\cos\frac{\pi}{7}.\cos\frac{2\pi}{7}.\cos\frac{4\pi}{7}\\ =-2\sin\frac{\pi}{7}.\cos\frac{\pi}{7}.\cos\frac{2\pi}{7}.\cos\frac{4\pi}{7}.\frac1{2\sin\frac{\pi}{7}}\\ =-\sin\frac{2\pi}{7}.\cos\frac{2\pi}{7}.\cos\frac{4\pi}{7}.\frac1{2\sin\frac{\pi}{7}}\\ =-2\sin\frac{2\pi}{7}.\cos\frac{2\pi}{7}.\cos\frac{4\pi}{7}.\frac1{4\sin\frac{\pi}{7}}\\ =-\sin\frac{4\pi}{7}.\cos\frac{4\pi}{7}.\frac{1}{4\sin\frac{\pi}{7}}\\ =-2\sin\frac{4\pi}{7}.\cos\frac{4\pi}{7}.\frac{1}{8\sin\frac{\pi}{7}}\\ =\sin\frac{8\pi}{7}.\frac{1}{8\sin\frac{\pi}{7}}=-\sin\left(\pi-\frac{8\pi}{7}\right).\frac{1}{8\sin\frac{\pi}{7}}\\ =-\sin\left(-\frac{\pi}{7}\right).\frac{1}{8\sin\frac{\pi}{7}}=\sin\frac{\pi}{7}.\frac{1}{8\sin\frac{\pi}{7}}= ...\\ \)