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$cos\frac{\pi}{7}+cos\frac{3\pi}{7}+cos\frac{5\pi}{7}=\\\\=\frac{1}{2sin\frac{\pi}{7}}\cdot (2sin\frac{\pi}{7}cos\frac{\pi}{7}+2sin\frac{\pi}{7}cos\frac{3\pi}{7}+2sin\frac{\pi}{7}cos\frac{5\pi}{7})=\\\\=\frac{1}{2sin\frac{\pi}{7}}\cdot (sin\frac{2\pi}{7}+2\cdot \frac{1}{2}(sin\frac{4\pi}{7}+sin(-\frac{2\pi}{7}))+2\cdot \frac{1}{2}(sin\frac{6\pi}{7}+sin(-\frac{4\pi}{7}})))\\\\=\frac{1}{2sin\frac{\pi}{7}}(sin\frac{2\pi}{7}+sin\frac{4\pi}{7}-sin\frac{2\pi}{7}+sin\frac{6\pi}{7}-sin\frac{4\pi}{7})=$

$=\frac{sin\frac{6\pi}{7}}{2sin\frac{\pi}{7}}=\frac{sin(\pi -\frac{\pi}{7})}{2sin\frac{\pi}{7}}=\frac{sin\frac{\pi}{7}}{2sin\frac{\pi}{7}}=\frac{1}{2}$