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Question

In $$\triangle ABC$$, $$\displaystyle cosec\:\frac{A}{2}+cosec\:\frac{B}{2}+cosec\:\frac{C}{2}\geq m$$ Find m 


Solution

In $$\triangle ABC$$, we know that $$\displaystyle \sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\leq \frac{1}{8}$$
$$\Rightarrow $$   $$\displaystyle \frac{cosec\:\frac{A}{2}+cosec\:\frac{B}{2}+cosec\:\frac{C}{2}}{3}\geq \left ( cosec\:\frac{A}{2}cosec\:\frac{B}{2}cosec\:\frac{C}{2} \right )^{1/3}$$
$$\displaystyle \frac{cosec\:\frac{A}{2}+cosec\:\frac{B}{2}+cosec\:\frac{C}{2}}{3}\geq \left ( \frac{1}{\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}} \right )^{1/3}$$
$$\displaystyle \frac{cosec\:\frac{A}{2}+cosec\:\frac{B}{2}+cosec\:\frac{C}{2}}{3}\geq \left ( 8 \right )^{1/3}$$
$$\displaystyle cosec\:\frac{A}{2}+cosec\:\frac{B}{2}+cosec\:\frac{C}{2}\geq 6$$
Ans: m=6

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