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Ta có:
$\cos A+\cos B=2\cos(\dfrac{A}{2}+\dfrac{B}{2})\cos(\dfrac{A}{2}-\dfrac{B}{2})\le2\cos(\dfrac{A}{2}+\dfrac{B}{2})$
$\cos C+\cos\dfrac{\pi}{3}=2\cos(\dfrac{C}{2}+\dfrac{\pi}{6})\cos(\dfrac{C}{2}-\dfrac{\pi}{6})\le2\cos(\dfrac{C}{2}+\dfrac{\pi}{6})$
$\cos(\dfrac{A}{2}+\dfrac{B}{2})+\cos(\dfrac{C}{2}+\dfrac{\pi}{6})=2\cos\dfrac{\pi}{3}\cos(\dfrac{A}{4}+\dfrac{B}{4}-\dfrac{C}{4}-\dfrac{\pi}{12})\le2\cos\dfrac{\pi}{3}$
Suy ra: $\cos A+\cos B+\cos C\le3\cos\dfrac{\pi}{3}=\dfrac{3}{2}$
và $\cos A\cos B\cos C \le \left ( \dfrac{\cos A+\cos B+\cos C}{3} \right )^3\le\dfrac{1}{8}$
Ta có:
$\left(1+\dfrac{1}{\cos A}\right)\left(1+\dfrac{1}{\cos B}\right)\left(1+\dfrac{1}{\cos C}\right)$
$=1+\left(\dfrac{1}{\cos A}+\dfrac{1}{\cos B}+\dfrac{1}{\cos C}\right)+\left(\dfrac{1}{\cos A\cos B}+\dfrac{1}{\cos B\cos C}+\dfrac{1}{\cos C\cos A}\right)+\dfrac{1}{\cos A\cos B\cos C}$
$\ge1+\dfrac{3}{\sqrt[3]{\cos A\cos B\cos C}}+\dfrac{3}{\sqrt[3]{\cos^2A\cos^2B\cos^2C}}+8\ge27$
Dấu bằng xảy ra khi: $\Delta ABC$ đều.
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