Question

If $A,B,C$ are angles of a triangle and angle $A$ is obtuse, then the exhaustive set of value of $\tan B\tan C$ is

• A. $(0,1)$
• B. $(0,2)$
• C. $(0,0.5)$
• D. any positive real number

Answer 1

The correct option is A $(0,1)$

$A+B+C=\pi B+C=\pi -A\tan A=-\tan (B+C)\tan A=\frac{\tan B+\tan C}{\tan B\tan C-1}$
As angle $A$ is obtuse angle and rest other angles are acute, so
$\tan A<0,\tan B>0,\tan C>0\Rightarrow \frac{\tan B+\tan C}{\tan B\tan C-1}<0$
$\Rightarrow \tan B\tan C<1$
Therefore, $\tan B\tan C\in (0,1)$