Lucille Cummings

2022-06-26

Let $\mathrm{△}ABC$ be an acute-angled triangle. Prove that $\sum _{\text{cyc}}\sqrt{\mathrm{cot}A+\mathrm{cot}B}\ge 2\sqrt{2}$

last99erib

Expert

hint:
$\sqrt{\mathrm{cot}A+\mathrm{cot}B}=\sqrt{\frac{sin\left(A+B\right)}{sinAsinB}}=\sqrt{\frac{sinC}{sinAsinB}}=\frac{sinC}{\sqrt{sinAsinBsinC}}$
$3\sqrt[3]{sinAsinBsinC}\le sinA+sinB+sinC\le \frac{3\sqrt{3}}{2}$

Do you have a similar question?