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    text  If  end text fraction numerator cos begin display style text end text end style A over denominator cos begin display style text end text end style B end fraction equals n text  and  end text fraction numerator sin begin display style text end text end style A over denominator sin begin display style text end text end style B end fraction equals m text  then  end text open parentheses m squared minus n squared close parentheses sin squared text end text B equals

    1. 1 minus n squared
    2. 1 plus n squared
    3. 1 minus n
    4. 1 plus n

    hintHint:

    In this question first we have to do some simple algebra then we will use the formula of a squared minus b squared equals left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis to simplify the expression. Then we have to use the formulas of sin open parentheses A minus B close parentheses equals sin open parentheses A close parentheses cos open parentheses B close parentheses minus cos open parentheses A close parentheses sin open parentheses B close parentheses and sin open parentheses A plus B close parentheses equals sin open parentheses A close parentheses cos open parentheses B close parentheses plus cos open parentheses A close parentheses sin open parentheses B close parentheses . After that we will use the formula of sin open parentheses alpha close parentheses sin open parentheses beta close parentheses equals 1 half left parenthesis cos open parentheses alpha minus beta close parentheses minus cos open parentheses alpha plus beta close parentheses right parenthesis then cos open parentheses 2 alpha close parentheses equals 2 cos open parentheses alpha close parentheses squared minus 1 to get the final answer.

    The correct answer is: 1 minus n squared

      In this question we are given expression fraction numerator cos begin display style text end text end style A over denominator cos begin display style text end text end style B end fraction equals n space and fraction numerator sin begin display style text end text end style A over denominator sin begin display style text end text end style B end fraction equals m and We have to find value of open parentheses m squared minus n squared close parentheses sin squared text end text B
      Step1: Finding the value of open parentheses m squared minus n squared close parentheses
      We know that fraction numerator sin begin display style text end text end style A over denominator sin begin display style text end text end style B end fraction equals m and fraction numerator cos begin display style text end text end style A over denominator cos begin display style text end text end style B end fraction equals n space
      m squared minus n squared equals left parenthesis fraction numerator sin open parentheses A close parentheses over denominator sin open parentheses B close parentheses end fraction right parenthesis squared minus left parenthesis fraction numerator cos open parentheses A close parentheses over denominator cos open parentheses B close parentheses end fraction right parenthesis squared
      =>fraction numerator sin open parentheses A close parentheses squared cos open parentheses B close parentheses squared minus sin open parentheses B close parentheses squared cos open parentheses A close parentheses squared over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction
      Step2: Using the formula of a squared minus b squared equals left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis we get,
      fraction numerator left parenthesis sin open parentheses A close parentheses cos open parentheses B close parentheses right parenthesis squared minus left parenthesis sin open parentheses B close parentheses cos open parentheses A close parentheses right parenthesis squared over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction
      =>fraction numerator left parenthesis sin open parentheses A close parentheses cos open parentheses B close parentheses minus sin open parentheses B close parentheses cos open parentheses A close parentheses right parenthesis left parenthesis sin open parentheses A close parentheses cos open parentheses B close parentheses plus sin open parentheses B close parentheses cos open parentheses A close parentheses right parenthesis over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction
      Step3: Using the formula of sin open parentheses A minus B close parentheses equals sin open parentheses A close parentheses cos open parentheses B close parentheses minus cos open parentheses A close parentheses sin open parentheses B close parentheses and sin open parentheses A plus B close parentheses equals sin open parentheses A close parentheses cos open parentheses B close parentheses plus cos open parentheses A close parentheses sin open parentheses B close parentheses we get,
      =>fraction numerator sin open parentheses A minus B close parentheses sin open parentheses A plus B close parentheses over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction
      Step4: By using the formula sin open parentheses alpha close parentheses sin open parentheses beta close parentheses equals 1 half left parenthesis cos open parentheses alpha minus beta close parentheses minus cos open parentheses alpha plus beta close parentheses right parenthesis
      =>1 half open parentheses fraction numerator cos open parentheses left parenthesis A plus B right parenthesis minus left parenthesis A minus B right parenthesis close parentheses minus cos open parentheses left parenthesis A plus B right parenthesis plus left parenthesis A minus B right parenthesis close parentheses over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction close parentheses
      =>1 half open parentheses fraction numerator cos open parentheses 2 B close parentheses minus cos open parentheses 2 A close parentheses over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction close parentheses
      Step5: By using the identity cos open parentheses 2 alpha close parentheses equals 2 cos open parentheses alpha close parentheses squared minus 1
      1 half open parentheses fraction numerator left parenthesis 2 cos open parentheses B close parentheses squared minus 1 right parenthesis minus left parenthesis 2 cos open parentheses A close parentheses squared minus 1 right parenthesis over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction close parentheses
      =>open parentheses fraction numerator cos open parentheses B close parentheses squared minus cos open parentheses A close parentheses squared over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction close parentheses
      By simplify it we get,
      =>fraction numerator cos open parentheses B close parentheses squared over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction minus fraction numerator cos open parentheses A close parentheses squared over denominator sin open parentheses B close parentheses squared cos open parentheses B close parentheses squared end fraction
      =>m squared minus n squared equals fraction numerator 1 over denominator sin open parentheses B close parentheses squared end fraction minus fraction numerator n squared over denominator sin open parentheses B close parentheses squared end fraction
      By cross multiplying terms we get,
      left parenthesis m squared minus n squared right parenthesis sin open parentheses B close parentheses squared equals 1 minus n squared.

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