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    Maths-
    General
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    Question

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

    1. left parenthesis a plus b plus c right parenthesis divided by 4 capital delta
    2. left parenthesis a squared plus b squared plus c squared right parenthesis divided by 4 capital delta
    3. left parenthesis left parenthesis a plus b plus c right parenthesis squared right parenthesis divided by 4 capital delta
    4. left parenthesis left parenthesis a plus b plus c right parenthesis squared right parenthesis divided by 2 capital delta

    hintHint:

    In this question we have to use the cosine rule and sine rule.
    First write all the equations of cosine rule then add them and replace the product of the sides from given formulas.a squared equals space b squared plus c squared minus 2 b c cos open parentheses A close parentheses b squared equals a squared plus c squared minus 2 a c cos open parentheses B close parentheses c squared equals a squared plus b squared minus 2 a b cos open parentheses C close parentheses , 1 half a b sin open parentheses C close parentheses equals increment 1 half a c sin open parentheses B close parentheses equals increment 1 half b c sin open parentheses A close parentheses equals increment

    The correct answer is: left parenthesis a squared plus b squared plus c squared right parenthesis divided by 4 capital delta

      In this question we have to find the value of cot open parentheses A close parentheses plus cot open parentheses B close parentheses plus cot open parentheses C close parentheses
      We know that in a triangle
      1 half a b sin open parentheses C close parentheses equals increment 1 half a c sin open parentheses B close parentheses equals increment 1 half b c sin open parentheses A close parentheses equals increment
      By Using Cosine Rule
      a squared equals space b squared plus c squared minus 2 b c cos open parentheses A close parentheses b squared equals a squared plus c squared minus 2 a c cos open parentheses B close parentheses c squared equals a squared plus b squared minus 2 a b cos open parentheses C close parentheses
      By adding above three equations we get,
      a squared plus b squared plus c squared equals 2 b c cos open parentheses A close parentheses plus 2 a c cos open parentheses B close parentheses plus 2 a b cos open parentheses C close parentheses
      We can replace b c comma a c comma a b by fraction numerator 2 increment over denominator sin open parentheses A close parentheses end fraction comma fraction numerator 2 increment over denominator sin open parentheses B close parentheses end fraction comma fraction numerator 2 increment over denominator sin open parentheses C close parentheses end fraction
      By substituting we get,
      a squared plus b squared plus c squared equals fraction numerator 4 increment cos open parentheses A close parentheses over denominator sin open parentheses A close parentheses end fraction plus fraction numerator 4 increment cos open parentheses B close parentheses over denominator sin open parentheses B close parentheses end fraction plus fraction numerator 4 increment cos open parentheses C close parentheses over denominator sin open parentheses C close parentheses end fraction
      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
      =>fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction equals cot open parentheses A close parentheses plus cot open parentheses B close parentheses plus cot open parentheses C close parentheses

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