Question

$\red {16}$
$\frac{cos\frac{A}{2} cos\frac{B}{2} cos\frac{C}{2} }{\left(1-sin\frac{A}{2}\right)\left(1-sin\frac{B}{2}\right)\left(1-sin\frac{C}{2}\right)}≥3\sqrt{3}$​

Answer 1

Hey there!

$Let \ f(x) = \log \left(\dfrac{\cos x}{1- \sin x}\right), f'(x) = -\dfrac{1}{\cos x}, f''(x) = -\dfrac{\sin x}{\cos^2 x} < 0$

$\displaystyle \sum f\left(\dfrac{A}{2}\right) \geq 3f\left(\dfrac{A+B+C}{6}\right) = 3 \log \sqrt{3}$

$\displaystyle \sum \log \left( \dfrac{\cos \dfrac{A}{2}}{1-\sin \dfrac{A}{2}}\right) \geq\log(\sqrt{3})^2$

$\displaystyle \prod \left( \dfrac{\cos \dfrac{A}{2}}{1-\sin \dfrac{A}{2}}\right) \geq 3\sqrt{3}$

Hence, we can say that,

$\dfrac{\cos \dfrac{A}{2}.\cos\dfrac{B}{2}.\cos\dfrac{C}{2}}{\left(1 - \sin \dfrac{A}{2}\right)\left(1 - \sin \dfrac{B}{2}\right)\left(1 - \sin \dfrac{C}{2}\right)} \geq 3\sqrt{3}$

Answer 2

Answer:

• pls mark as brainliest