If A + B + C = ,(A,B,C > 0) and the angle C is obtuse then - Vidyadip

MCQ

If $A + B + C = \pi \,(A,B,C > 0)$ and the angle $C$ is obtuse then

A
$\tan A\,\tan B > 1$
✓
$\tan A\,\tan B < 1$
C
$\tan A\,\,\tan B = 1$
D
None of these

✓

Answer

Correct option: B.

$\tan A\,\tan B < 1$

b
(b) $A + B + C = \pi \Rightarrow A + B = \pi - C$

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

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

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

Now $C$ is an obtuse angle, hence

$ \Rightarrow \tan C < 0 \Rightarrow - \tan C > 0$

$ \Rightarrow \frac{{\tan A + \tan B}}{{1 - \tan A\tan B}} > 0$

$\Rightarrow 1 - \tan A\tan B > 0$

$(\because A,B$ are acute angles; $\therefore \tan A > 0,\tan B > 0 )$

$ \Rightarrow \tan A\tan B < 1$.

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