Вычислите выражение: [latex]cos \pi /7 + cos3 \pi /7 + cos5 \pi /7[/latex] .

[latex]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})=[/latex] [latex]=\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}[/latex]