Обчислити 8/(√3)*cos10°*sin20°*sin40°

[latex] \frac{8}{\sqrt3} \cdot cos10\cdot sin20\cdot sin40=\\\\=[\, sin \alpha \cdot sin \beta =\frac{1}{2}(cos( \alpha - \beta ) -cos (\alpha + \beta ))\, ]=\\\\=\frac{8}{\sqrt3} \cdot cos10\cdot \frac{1}{2}(cos20-cos60)= \\\\=\frac{4}{\sqrt3}\cdot cos10\cdot (cos20-\frac{1}{2})=\\\\=\frac{4}{\sqrt3} \cdot cos10\cdot cos20- \frac{4}{\sqrt3} cos10\cdot \frac{1}{2}=\\\\=\frac{4}{\sqrt3}\cdot \frac{1}{2}(cos10+cos30)- \frac{2}{\sqrt3} \cdot cos10=[/latex] [latex]= \frac{2}{\sqrt3} \cdot cos10+\frac{2}{\sqrt3}\cdot cos30-\frac{2}{\sqrt3}cos10=[/latex] [latex]=\frac{2}{\sqrt3}\cdot cos30=\frac{2}{\sqrt3}\cdot \frac{\sqrt3}{2}=1[/latex]