\(cos\frac{\pi}{4}=2cos^2\frac{\pi}{8}-1\Rightarrow cos^2\frac{\pi}{8}=\frac{cos\frac{\pi}{4}+1}{2}\)
\(\Rightarrow cos^2\frac{\pi}{8}=\frac{2+\sqrt{2}}{4}\Rightarrow cos\frac{\pi}{8}=\frac{\sqrt{2+\sqrt{2}}}{2}\) (do \(0< \frac{\pi}{8}< \frac{\pi}{2}\) nên \(cos\frac{\pi}{8}>0\))
\(M=cos\frac{\pi}{7}-cos\frac{2\pi}{7}+cos\frac{3\pi}{7}\)
\(\Rightarrow2M.sin\frac{\pi}{7}=2sin\frac{\pi}{7}cos\frac{\pi}{7}-2sin\frac{\pi}{7}cos\frac{2\pi}{7}+2sin\frac{\pi}{7}cos\frac{3\pi}{7}\)
\(=sin\frac{2\pi}{7}-sin\frac{3\pi}{7}+sin\frac{\pi}{7}+sin\frac{4\pi}{7}-sin\frac{2\pi}{7}\)
\(=-sin\frac{3\pi}{7}+sin\frac{\pi}{7}+sin\left(\pi-\frac{3\pi}{7}\right)\)
\(=-sin\frac{3\pi}{7}+sin\frac{\pi}{7}+sin\frac{3\pi}{7}=sin\frac{\pi}{7}\)
\(\Rightarrow M=\frac{sin\frac{\pi}{7}}{2sin\frac{\pi}{7}}=\frac{1}{2}\)