[tex]\red {16}[/tex]
[tex]\frac{cos\frac{A}{2} cos\frac{B}{2} cos\frac{C}{2} }{\left(1-sin\frac{A}{2}\right)\left(1-sin\frac{B}{2}\right)\left(1-sin\frac{C}{2}\right)}≥3\sqrt{3}[/tex]
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[tex]\red {16}[/tex]
[tex]\frac{cos\frac{A}{2} cos\frac{B}{2} cos\frac{C}{2} }{\left(1-sin\frac{A}{2}\right)\left(1-sin\frac{B}{2}\right)\left(1-sin\frac{C}{2}\right)}≥3\sqrt{3}[/tex]
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[tex]Let \ f(x) = \log \left(\dfrac{\cos x}{1- \sin x}\right), f'(x) = -\dfrac{1}{\cos x}, f''(x) = -\dfrac{\sin x}{\cos^2 x} < 0[/tex]
[tex]\displaystyle \sum f\left(\dfrac{A}{2}\right) \geq 3f\left(\dfrac{A+B+C}{6}\right) = 3 \log \sqrt{3}[/tex]
[tex]\displaystyle \sum \log \left( \dfrac{\cos \dfrac{A}{2}}{1-\sin \dfrac{A}{2}}\right) \geq\log(\sqrt{3})^2[/tex]
[tex]\displaystyle \prod \left( \dfrac{\cos \dfrac{A}{2}}{1-\sin \dfrac{A}{2}}\right) \geq 3\sqrt{3}[/tex]
Hence, we can say that,
[tex]\dfrac{\cos \dfrac{A}{2}.\cos\dfrac{B}{2}.\cos\dfrac{C}{2}}{\left(1 - \sin \dfrac{A}{2}\right)\left(1 - \sin \dfrac{B}{2}\right)\left(1 - \sin \dfrac{C}{2}\right)} \geq 3\sqrt{3}[/tex]
Answer: