Question

In $△ABC$, $cosec\frac{A}{2}+cosec\frac{B}{2}+cosec\frac{C}{2}\ge m$ Find m

Answer 1

In $△ABC$, we know that $\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\le \frac{1}{8}$

$\Rightarrow$ $\frac{cosec\frac{A}{2}+cosec\frac{B}{2}+cosec\frac{C}{2}}{3}\ge {\left(cosec\frac{A}{2}cosec\frac{B}{2}cosec\frac{C}{2}\right)}^{1/3}$

$\frac{cosec\frac{A}{2}+cosec\frac{B}{2}+cosec\frac{C}{2}}{3}\ge {\left(\frac{1}{\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}\right)}^{1/3}$

$\frac{cosec\frac{A}{2}+cosec\frac{B}{2}+cosec\frac{C}{2}}{3}\ge {\left(8\right)}^{1/3}$

$cosec\frac{A}{2}+cosec\frac{B}{2}+cosec\frac{C}{2}\ge 6$
Ans: m=6