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

[latex]cosfrac{pi}{7}+cosfrac{3pi}{7}+cosfrac{5pi}{7}=\\=frac{1}{2sinfrac{pi}{7}}cdot (2sinfrac{pi}{7}cosfrac{pi}{7}+2sinfrac{pi}{7}cosfrac{3pi}{7}+2sinfrac{pi}{7}cosfrac{5pi}{7})=\\=frac{1}{2sinfrac{pi}{7}}cdot (sinfrac{2pi}{7}+2cdot frac{1}{2}(sinfrac{4pi}{7}+sin(-frac{2pi}{7}))+2cdot frac{1}{2}(sinfrac{6pi}{7}+sin(-frac{4pi}{7}})))\\=frac{1}{2sinfrac{pi}{7}}(sinfrac{2pi}{7}+sinfrac{4pi}{7}-sinfrac{2pi}{7}+sinfrac{6pi}{7}-sinfrac{4pi}{7})=[/latex][latex]=frac{sinfrac{6pi}{7}}{2sinfrac{pi}{7}}=frac{sin(pi -frac{pi}{7})}{2sinfrac{pi}{7}}=frac{sinfrac{pi}{7}}{2sinfrac{pi}{7}}=frac{1}{2}[/latex]