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    Maths-
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    Question

    Let alpha blank a n d blank beta be such that pi less than alpha minus beta less than 3 pi. If sin invisible function application alpha plus sin invisible function application beta equals negative fraction numerator 21 over denominator 65 end fraction blank a n d cos invisible function application alpha plus cos invisible function application beta equals negative fraction numerator 17 over denominator 65 end fraction comma then the value of cos invisible function application fraction numerator alpha minus beta over denominator 2 end fraction is

    1. negative fraction numerator 3 over denominator square root of 130 end fraction  
    2. fraction numerator 3 over denominator square root of 130 end fraction  
    3. fraction numerator 6 over denominator 65 end fraction  
    4. negative fraction numerator 6 over denominator 65 end fraction  

    The correct answer is: negative fraction numerator 3 over denominator square root of 130 end fraction

      W e blank h a v e sin invisible function application alpha plus sin invisible function application beta equals negative fraction numerator 21 over denominator 65 end fraction blank
      cos invisible function application alpha plus cos invisible function application beta equals negative fraction numerator 17 over denominator 65 end fraction
      Squaring Eq. (i), we get sin to the power of 2 end exponent invisible function application alpha plus sin to the power of 2 end exponent invisible function application beta plus 2 sin invisible function application alpha sin invisible function application beta equals open parentheses fraction numerator 21 over denominator 65 end fraction close parentheses to the power of 2 end exponent
      Squaring Eq. (ii), we get cos to the power of 2 end exponent invisible function application alpha plus cos to the power of 2 end exponent invisible function application beta plus 2 cos invisible function application alpha cos invisible function application beta equals open parentheses fraction numerator 27 over denominator 65 end fraction close parentheses to the power of 2 end exponent
      Adding Eqs. (iii) and (iv), we get 2 plus 2 cos invisible function application open parentheses alpha minus beta close parentheses equals fraction numerator 1 over denominator open parentheses 65 close parentheses to the power of 2 end exponent end fraction open square brackets open parentheses 27 close parentheses to the power of 2 end exponent plus open parentheses 21 close parentheses to the power of 2 end exponent close square brackets equals fraction numerator 1 over denominator open parentheses 65 close parentheses to the power of 2 end exponent end fraction open parentheses 729 plus 441 close parentheses
      rightwards double arrow 2 plus 2 blank cos invisible function application open parentheses alpha minus beta close parentheses equals fraction numerator 1 over denominator open parentheses 65 close parentheses to the power of 2 end exponent end fraction open parentheses 1170 close parentheses equals fraction numerator 18 over denominator 65 end fraction
      rightwards double arrow 1 plus cos invisible function application open parentheses alpha minus beta close parentheses equals fraction numerator 9 over denominator 65 end fraction
      rightwards double arrow 2 cos to the power of 2 end exponent invisible function application fraction numerator alpha minus beta over denominator 2 end fraction equals fraction numerator 9 over denominator 65 end fraction
      rightwards double arrow cos invisible function application fraction numerator alpha minus beta over denominator 2 end fraction equals negative fraction numerator 3 over denominator square root of 130 end fraction blank open square brackets because pi less than alpha minus beta less than 3 pi rightwards double arrow fraction numerator pi over denominator 2 end fraction less than fraction numerator alpha minus beta over denominator 2 end fraction less than fraction numerator 3 pi over denominator 2 end fraction rightwards double arrow cos invisible function application open parentheses fraction numerator alpha minus beta over denominator 2 end fraction close parentheses less than 0 close square brackets

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