Question

Which of the following inequalities holds true in any $△$ $ABC$?

• A. $\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\le \frac{1}{8}$
• B. $\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\le \frac{3\sqrt{3}}{8}$
• C. ${\sin }^2\frac{B}{2}+{\sin }^2\frac{B}{2}+{\sin }^2\frac{C}{2}<\frac{3}{4}$
• D. ${\cos }^2\frac{A}{2}+{\cos }^2\frac{B}{2}+{\cos }^2\frac{C}{2}\le \frac{9}{4}$

Answer 1

Answer: (A), (B), (D)

$\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\le \frac{1}{8}$
$\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\le \frac{3\sqrt{3}}{8}$
${\cos }^2\frac{A}{2}+{\cos }^2\frac{B}{2}+{\cos }^2\frac{C}{2}\le \frac{9}{4}$

For part A and B refer above formulas,
part C,
${\sin }^2\frac{A}{2}+{\sin }^2\frac{B}{2}+{\sin }^2\frac{C}{2}$
$=\frac{1-\cos A}{2}+\frac{1-\cos B}{2}+\frac{1-\cos C}{2}$
$=\frac{3}{2}-\frac{1}{2}(\cos A+\cos B+\cos C)$
$\ge \frac{3}{2}-\frac{1}{2}.\frac{3}{2}=\frac{3}{4}$
so option C is incorrect
Now option D
${\cos }^2\frac{A}{2}+{\cos }^2\frac{B}{2}+{\cos }^2\frac{C}{2}$
$=\frac{1+\cos A}{2}+\frac{1+\cos B}{2}+\frac{1+\cos C}{2}$
$=\frac{3}{2}+\frac{1}{2}(\cos A+\cos B+\cos C)$
$\le \frac{3}{2}+\frac{1}{2}.\frac{3}{2}=\frac{9}{4}$