Question Number 170115 by Shrinava last updated on 16/May/22
$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{then}: \\ $$$$\frac{\mathrm{cos}\:\frac{\mathrm{A}}{\mathrm{2}}\:\centerdot\:\mathrm{cos}\:\frac{\mathrm{B}}{\mathrm{2}}\:\centerdot\:\mathrm{cos}\:\frac{\mathrm{C}}{\mathrm{2}}}{\left(\mathrm{1}\:-\:\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}\right)\centerdot\left(\mathrm{1}\:-\:\mathrm{sin}\:\frac{\mathrm{B}}{\mathrm{2}}\right)\centerdot\left(\mathrm{1}\:-\:\mathrm{sin}\:\frac{\mathrm{C}}{\mathrm{2}}\right)}\:\geqslant\:\mathrm{3}\sqrt{\mathrm{3}} \\ $$