Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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If Sin (B + C − A), Sin (C + a − B), Sin (A + B − C) Are in A.P., Then Cot A, Cot B and Cot C Are in - Mathematics

MCQ
Sum

If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in

Options

  • GP

  • HP

  • AP

  • None of these

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Solution

HP
Given:
sin (B + C − A), sin (C + A − B) and sin (A + B − C) are in A.P.
\[\Rightarrow \sin\left( C + A - B \right) - \sin\left( B + C - A \right) = \sin\left( A + B - C \right) - \sin\left( C + A - B \right)\]
\[ \Rightarrow 2\sin\left( \frac{C + A - B - B - C + A}{2} \right) \cos\left( \frac{C + A - B + B + C - A}{2} \right) = 2\sin\left( \frac{A + B - C - C - A + B}{2} \right) \cos\left( \frac{A + B - C + C + A - B}{2} \right)\]
\[ \Rightarrow \sin\left( A - B \right) \cos C = \sin\left( B - C \right) \cos A\]
\[ \Rightarrow \sin A \cos B \cos C - \cos A \sin B \cos C = \sin B \cos C\cos A - \cos B \sin C \cos A\]
\[ \Rightarrow 2\sin B \cos A \cos C = \sin A \cos B \cos C + \cos A \cos B \sin C\] 
Dividing both sides by cosA cosB cosC:
\[2\tan B = \tan A + \tan C \]
\[ \Rightarrow \frac{2}{cotB} = \frac{1}{cotA} + \frac{1}{cotC}\]

Hence, cotA, cotB and cotC are in HP.

Concept: Transformation Formulae
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 8 Transformation formulae
Q 12 | Page 22
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