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# If in a δ a B C , Tan a + Tan B + Tan C = 0 , Then Cot a Cot B Cot C = - Mathematics

MCQ

If in a  $∆ ABC, \tan A + \tan B + \tan C = 0$, then

$\cot A \cot B \cot C =$

#### Options

• 6

• 1

• $\frac{1}{6}$

•  none of these

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#### Solution

none of these
ABC is a triangle.

$\therefore A + B + C = \pi$
$\Rightarrow A + B = \pi - C$
$\Rightarrow \tan\left( A + B \right) = \tan\left( \pi - C \right)$
$\Rightarrow \frac{\text{ tan } A + \text{ tan } B}{1 - \text{ tan } A \text{ tan } B} = - \text{ tan } C$
$\Rightarrow \text{ tan } A + \text{ tan } B = - \text{ tan } C + \text{ tan } A \text{ tan } B \text{ tan } C$
$\Rightarrow \text{ tan } A + \text{ tan } B + \text{ tan } C = \text{ tan } A \text{ tan } B \text{ tan } C$
$\Rightarrow 0 = \text{ tan } A \text{ tan } B \text{ tan } C [Given: \text{ tan } A \text{ tan } B \text{ tan } C = 0]$
$\Rightarrow \text{ tan } A \text{ tan } B \text{ tan } C = 0$
$\Rightarrow \frac{1}{\text{ tan } A \text{ tan } B \text{ tan }C} = \frac{1}{0}$
$\Rightarrow \text{ cot } A \text{ cot } B \text{ cot } C \to \infty$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Q 7 | Page 43
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