Chứng minh $cosA+cosB+cosC\leq 1+\frac{cos^{2}\frac{B-C}{2}}{2}$
Chứng minh $cosA+cosB+cosC\leq 1+\frac{cos^{2}\frac{B-C}{2}}{2}$
Chứng minh $cosA+cosB+cosC\leq 1+\frac{cos^{2}\frac{B-C}{2}}{2}$
BĐT $\Leftrightarrow 1+\frac{\cos^2\frac{B-C}{2}}{2}-\cos A-(\cos B+ \cos C) \geqslant 0$
$\Leftrightarrow 1+\frac{\cos^2\frac{B-C}{2}}{2}-(1-2\sin^2\frac{A}{2})-2 \cos \frac{B+C}{2} \cos \frac{B-C}{2}\geqslant 0$
$\Leftrightarrow \frac{\cos^2\frac{B-C}{2}}{2}+2\sin^2\frac{A}{2}-2 \sin \frac{A}{2} \cos \frac{B-C}{2}\geqslant 0$
$\Leftrightarrow (\cos\frac{B-C}{2}-2\sin \frac{A}{2})^2 \geqslant 0$
Đẳng thức xảy ra khi $A=B=C=60^0$
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