Question

If $A,B,C$ are acute positive angles such that $A+B+C=\pi$ and $\cot A$ cot$B$ cot$C=K$, then

• A. $K\le \frac{1}{3\sqrt{3}}$
• B. $K\ge \frac{1}{3\sqrt{3}}$
• C. $K<\frac{1}{9}$
• D. $K>\frac{1}{3}$

Answer 1

The correct option is A $K\le \frac{1}{3\sqrt{3}}$

We have,

$\tan (A+B+C)=\frac{(\tan A+\tan B+\tan C-\tan A\tan B\tan C)}{(1-\tan A\tan B-\tan B\tan C-\tan C\tan A)}$

$A+B+C=\pi$

$\Rightarrow \tan (A+B+C)=\tan \pi =0$

$\Rightarrow \tan A+\tan B+\tan C=\tan A\tan B\tan C$ ....$\left(1\right)$
As all angles are acute, we can apply the $A.M.\ge G.M.$ inequality for $\tan A,\tan B,\tan C$

$\frac{\tan A+\tan B+\tan C}{3}\ge (\tan A\tan B\tan C{)}^{1/3}$

Using equation $\left(1\right),$ and $\cot A\cot B\cot C=K$

$\Rightarrow \frac{1}{3K}\ge (K{)}^{-1/3}$

$\Rightarrow K\le \frac{1}{3\sqrt{3}}$

$\Rightarrow \cot A\cot B\cot C\le \frac{1}{3\sqrt{3}}$