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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 =

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Question

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

MCQ
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Solution

 none of these
ABC is a triangle. 

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

shaalaa.com
Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.5 [Page 43]

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R.D. Sharma Mathematics [English] Class 11
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.5 | Q 7 | Page 43

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