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    Maths-
    General
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    Question

    In triangle A B C comma left parenthesis left parenthesis b minus c right parenthesis divided by a right parenthesis sin invisible function application left parenthesis left parenthesis B plus C right parenthesis divided by 2 right parenthesis equals

    1. cos invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis
    2. sin invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis
    3. sin invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis
    4. cos invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis

    hintHint:

    In this question first we have to use Sine rule fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals k. Then we will use some trigonometric formula sin open parentheses C close parentheses minus sin open parentheses D close parentheses space equals space 2 cos open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses sin open parentheses fraction numerator C minus D over denominator 2 end fraction close parentheses and sin open parentheses 2 x close parentheses equals 2 sin open parentheses x close parentheses cos open parentheses x close parentheses and then we will do some rearranging of the expression.

    The correct answer is: sin invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis

      In this question we have to find the value of open parentheses fraction numerator b minus c over denominator a end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
      By using Sine Rule
      fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals k
      =>open parentheses fraction numerator k sin open parentheses B close parentheses minus k sin open parentheses C close parentheses over denominator k sin open parentheses A close parentheses end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
      => open parentheses fraction numerator k sin open parentheses B close parentheses minus k sin open parentheses C close parentheses over denominator k sin open parentheses A close parentheses end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
      By using trigonometric formula
      sin open parentheses C close parentheses minus sin open parentheses D close parentheses space equals space 2 cos open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses sin open parentheses fraction numerator C minus D over denominator 2 end fraction close parentheses and sin open parentheses 2 x close parentheses equals 2 sin open parentheses x close parentheses cos open parentheses x close parentheses
      => open parentheses fraction numerator 2 cos open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses over denominator sin open parentheses A close parentheses end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
      We know that A plus B plus C space equals space pi
      fraction numerator B plus C over denominator 2 end fraction space equals space pi over 2 minus A over 2
      sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals sin open parentheses straight pi over 2 minus A over 2 close parentheses
      =>cos open parentheses A over 2 close parentheses
      Similarly cos open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals sin open parentheses A over 2 close parentheses
      => open parentheses fraction numerator 2 sin open parentheses begin display style A over 2 end style close parentheses sin open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses over denominator 2 sin open parentheses begin display style A over 2 end style close parentheses cos open parentheses begin display style A over 2 end style close parentheses end fraction close parentheses cos open parentheses A over 2 close parentheses
      => sin open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses

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