Question

Assertion :If $\tan A+\tan B+\tan C=3\sqrt{3}$, then triangle is equilateral Reason: In $\Delta ABC$,$\tan A+\tan B+\tan C=\tan A\tan B\tan C$.

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

In a triangle $A+B+C=\pi$

$A+B=\pi -C$

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

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

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

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

For an equilateral triangle.
$A=B=C={60}^0$

Hence
$\tan A.\tan B.\tan C$

$=(\sqrt{3}{)}^3$

$=3\sqrt{3}$

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