Question

Assertion :1 . If A,B, C are the angles of a triangle such that angleAis obtuse then
tan B tan C>1. because- Reason: In any triangle $tanA=\frac{tanB+tanC}{tanBtanC-1}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Given, Angle A is obtuse.
$\Rightarrow$B,C will be acute angles.
Sum of the angles in a triangle $={180}^o$
$\angle A+\angle B+\angle C=180$
$tanA=tan(180-B-C)$
$tanA=-tan(B+C)$
$tanA=\frac{(tanB+tanC)}{(1-tanAtanB)}$
$\Rightarrow tanA\times (tanCtanB-1)=(tanB+tanC)$
$\Rightarrow (tanCtanB-1)=\frac{(tanB+tanC)}{tanA}$
since, $\angle A$ is obtuse,tanA will be negative and $\angle B,\angle C$ are acute,tanB,tanC will be positive.
$\Rightarrow tanBtanC-1<0$
$\therefore$The given assertion is false and Reason is true.
Option D is correct