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    Maths-
    General
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    Question

    In capital delta A B C comma a squared c o t invisible function application A plus b squared c o t invisible function application B plus C squared c o t invisible function application C equals

    1. capital delta
    2. 2capital delta
    3. 3capital delta
    4. 4capital delta

    hintHint:

    In this question we have to use Sine rule
    a space equals 2 R sin open parentheses A close parentheses b equals 2 R sin open parentheses B close parentheses c equals 2 R sin open parentheses C close parentheses
    then we will use trigonometric formula sin open parentheses A close parentheses plus sin open parentheses B close parentheses equals space 2 sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses and cos open parentheses A plus B close parentheses plus cos open parentheses A minus B close parentheses equals space 2 sin open parentheses A close parentheses sin open parentheses B close parentheses . Then at last we will use fraction numerator a b c over denominator R end fraction equals 4 increment.

    The correct answer is: 4capital delta

      sin open parentheses 2 x close parentheses equals 2 sin open parentheses x close parentheses cos open parentheses x close parenthesesIn this question we have to find the value of a squared cot open parentheses A close parentheses plus b squared cot open parentheses B close parentheses plus c squared cot open parentheses C close parentheses
      By using Sine Rule
      a space equals 2 R sin open parentheses A close parentheses b equals 2 R sin open parentheses B close parentheses c equals 2 R sin open parentheses C close parentheses
      by substituting these value in the given expression
      => left parenthesis 2 R sin open parentheses A close parentheses right parenthesis squared cot open parentheses A close parentheses plus left parenthesis 2 R sin open parentheses B close parentheses right parenthesis squared cot open parentheses B close parentheses plus left parenthesis 2 R sin open parentheses C close parentheses right parenthesis squared cot open parentheses C close parentheses
      => 4 R squared left parenthesis sin squared open parentheses A close parentheses cot open parentheses A close parentheses plus sin squared open parentheses B close parentheses cot open parentheses B close parentheses plus sin squared open parentheses C close parentheses cot open parentheses C close parentheses right parenthesis
      We know that cot open parentheses x close parentheses equals fraction numerator cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses end fraction and
      => 4 R squared left parenthesis sin open parentheses A close parentheses cos open parentheses A close parentheses plus sin open parentheses B close parentheses cos open parentheses B close parentheses plus sin open parentheses C close parentheses cos open parentheses C close parentheses right parenthesis
      => 2 R squared left parenthesis 2 sin open parentheses A close parentheses cos open parentheses A close parentheses plus 2 sin open parentheses B close parentheses cos open parentheses B close parentheses plus 2 sin open parentheses C close parentheses cos open parentheses C close parentheses right parenthesis
      =>2 R squared left parenthesis sin open parentheses 2 A close parentheses plus sin open parentheses 2 B close parentheses plus sin open parentheses 2 C close parentheses right parenthesis
      We know that sin open parentheses A close parentheses plus sin open parentheses B close parentheses equals space 2 sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses
      => 2 R squared left parenthesis 2 sin open parentheses A plus B close parentheses cos open parentheses A minus B close parentheses plus 2 sin open parentheses C close parentheses cos open parentheses C close parentheses right parenthesis
      =>A plus B equals pi minus C
      => 2 R squared left parenthesis 2 sin open parentheses pi minus C close parentheses cos open parentheses A minus B close parentheses plus 2 sin open parentheses C close parentheses cos open parentheses C close parentheses right parenthesis
      => 2 R squared left parenthesis 2 sin open parentheses C close parentheses cos open parentheses A minus B close parentheses plus 2 sin open parentheses C close parentheses cos open parentheses C close parentheses right parenthesis
      => 4 R squared sin open parentheses C close parentheses left parenthesis cos open parentheses A minus B close parentheses plus cos open parentheses C close parentheses right parenthesis
      => 4 R squared sin open parentheses C close parentheses left parenthesis cos open parentheses A minus B close parentheses plus cos open parentheses pi minus left parenthesis A plus B right parenthesis close parentheses right parenthesis
      => 4 R squared sin open parentheses C close parentheses left parenthesis cos open parentheses A minus B close parentheses plus cos open parentheses A plus B close parentheses right parenthesis
      We know that cos open parentheses A plus B close parentheses plus cos open parentheses A minus B close parentheses equals space 2 sin open parentheses A close parentheses sin open parentheses B close parentheses
      => 8 R squared sin open parentheses C close parentheses sin open parentheses B close parentheses sin open parentheses A close parentheses
      Using Sine Rule
      sin open parentheses A close parentheses equals fraction numerator a over denominator 2 R end fraction comma sin open parentheses B close parentheses equals fraction numerator a over denominator 2 R end fraction comma sin open parentheses C close parentheses equals fraction numerator c over denominator 2 R end fraction
      => 8 R squared fraction numerator a over denominator 2 R end fraction cross times fraction numerator b over denominator 2 R end fraction cross times fraction numerator c over denominator 2 R end fraction
      =>fraction numerator a b c over denominator R end fraction
      We know that fraction numerator a b c over denominator R end fraction equals 4 increment
      =>4 increment

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