Question

Statement (I): If $A+B+C=\pi (A,B,C>0)$ and the angle $C$ is obtuse then $\tan A\tan B<1$.
Statement (II): If $A,B,C$ are acute positive angles
such that $A+B+C=\pi$ and $\cot A\cot B\cot C=K$ then
$k\le \frac{1}{3\sqrt{3}}$ Which of the above statements is correct?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II

Answer 1

Answer: (C)

Both I and II

Statement I
$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
$\tan (\pi -C)=-\tan C=\frac{\tan A+\tan B}{1-\tan A\tan B}$
C is obtuse angle
$\tan C$ is -ve and
$A+B$ is acute angle
$\tan (A+B)$ is +ve
$\Rightarrow \tan A\tan B<1$ True

Statement II
$\tan (A+B+C)=0,$ if $A+B+C=\pi$
$\therefore \tan A+\tan B+\tan C=\tan A\tan B\tan C=\frac{1}{k}$ [as given] - 1
By AM-GM on this $\tan A,\tan B,\tan C$ three +ve number
$AM\ge GM$
$\frac{\tan A+\tan B+\tan C}{3}\ge (\tan A\tan B\tan C{)}^{\frac{1}{3}}$
from 1
$(\tan A\tan B\tan C{)}^{\frac{2}{3}}\ge 3$
$\tan A\tan B\tan C\ge 3\sqrt{3}$
$\Rightarrow k\le \frac{1}{3\sqrt{3}}$ True.