Question

Assertion :In a acute angle triangle $\tan A\tan B\tan C=1$. Reason: In obtuse angle triangle $\tan A\tan B\tan C<0$.

• 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 false but Reason is true

Answer 1

The correct option is D Assertion false but Reason is true

We know that

$\tan A.\tan B.\tan C=\tan A+\tan B+\tan C$

By using $AM\ge GM$

$\Rightarrow \frac{\tan A+\tan B+\tan C}{3}\ge \sqrt[3]{3\tan A\tan B\tan C}$

$\Rightarrow {\left(\frac{\tan A+\tan B+\tan C}{3}\right)}^2\ge \tan A\tan B\tan C$

$\Rightarrow {\left(\frac{\tan A+\tan B+\tan C}{3}\right)}^3\ge \tan A+\tan B+\tan C$

$\Rightarrow {(\tan A+\tan B+\tan C)}^2\ge 3^3$

$\tan A+\tan B+\tan C\ge 3^{3/2}$

$\tan A+\tan B+\tan C\ge 3\sqrt{3}$

$\tan A.\tan B.\tan C\ge 3\sqrt{3}$

Minimum value: $\tan A.\tan B.\tan C\ge 3\sqrt{3}$

Hence, assertion is wrong.