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Question

If fraction numerator sin to the power of 4 invisible function application A over denominator a end fraction plus fraction numerator cos to the power of 4 invisible function application A over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction then the value of fraction numerator sin to the power of 8 end exponent invisible function application A over denominator a to the power of 3 end exponent end fraction plus fraction numerator cos to the power of 8 end exponent invisible function application A over denominator b to the power of 3 end exponent end fraction is equal to

  1. fraction numerator 1 over denominator left parenthesis a plus b right parenthesis to the power of 3 end exponent end fraction    
  2. fraction numerator a to the power of 3 end exponent b to the power of 3 end exponent over denominator left parenthesis a plus b right parenthesis to the power of 3 end exponent end fraction    
  3. fraction numerator a to the power of 2 end exponent b to the power of 2 end exponent over denominator left parenthesis a plus b right parenthesis to the power of 2 end exponent end fraction    
  4. None of these    

The correct answer is: fraction numerator 1 over denominator left parenthesis a plus b right parenthesis to the power of 3 end exponent end fraction


    It is given that fraction numerator sin to the power of 4 end exponent invisible function application A over denominator a end fraction plus fraction numerator cos to the power of 4 end exponent invisible function application A over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction
    Þ fraction numerator left parenthesis 1 minus cos invisible function application 2 A right parenthesis to the power of 2 end exponent over denominator 4 a end fraction plus fraction numerator left parenthesis 1 plus cos invisible function application 2 A right parenthesis to the power of 2 end exponent over denominator 4 b end fraction equals fraction numerator 1 over denominator a plus b end fraction
    Þ b left parenthesis a plus b right parenthesis left parenthesis 1 minus 2 cos invisible function application 2 A plus cos to the power of 2 end exponent invisible function application 2 A right parenthesis
    plus a left parenthesis a plus b right parenthesis left parenthesis 1 plus 2 cos invisible function application 2 A plus cos to the power of 2 end exponent invisible function application 2 A right parenthesis equals 4 a b
    Þleft curly bracket b left parenthesis a plus b right parenthesis plus a left parenthesis a plus b right parenthesis right curly bracket cos to the power of 2 end exponent invisible function application 2 A plus 2 left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis cos invisible function application 2 A
    plus a left parenthesis a plus b right parenthesis plus b left parenthesis a plus b right parenthesis minus 4 a b equals 0
    Þ left parenthesis a plus b right parenthesis to the power of 2 end exponent cos to the power of 2 end exponent invisible function application 2 A plus 2 left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis cos invisible function application 2 A plus left parenthesis a minus b right parenthesis to the power of 2 end exponent equals 0
    Þ left curly bracket left parenthesis a plus b right parenthesis cos invisible function application 2 A plus left parenthesis a minus b right parenthesis right curly bracket to the power of 2 end exponent equals 0or cos invisible function application 2 A equals fraction numerator b minus a over denominator b plus a end fraction
    Hence, fraction numerator sin to the power of 8 end exponent invisible function application A over denominator a to the power of 3 end exponent end fraction plus fraction numerator cos to the power of 8 end exponent invisible function application A over denominator b to the power of 3 end exponent end fraction equals fraction numerator left parenthesis 1 minus cos invisible function application 2 A right parenthesis to the power of 4 end exponent over denominator 16 a to the power of 3 end exponent end fraction plus fraction numerator left parenthesis 1 plus cos invisible function application 2 A right parenthesis to the power of 4 end exponent over denominator 16 b to the power of 3 end exponent end fraction
    equals fraction numerator 1 over denominator 16 a to the power of 3 end exponent end fraction open square brackets 1 minus fraction numerator b minus a over denominator b plus a end fraction close square brackets to the power of 4 end exponent plus fraction numerator 1 over denominator 16 b to the power of 3 end exponent end fraction open square brackets 1 plus fraction numerator b minus a over denominator b plus a end fraction close square brackets to the power of 4 end exponent
    equals fraction numerator 16 a to the power of 4 end exponent over denominator 16 a to the power of 3 end exponent left parenthesis b plus a right parenthesis to the power of 4 end exponent end fraction plus fraction numerator 16 b to the power of 4 end exponent over denominator 16 b to the power of 3 end exponent left parenthesis b plus a right parenthesis to the power of 4 end exponent end fraction
    equals fraction numerator 1 over denominator left parenthesis b plus a right parenthesis to the power of 4 end exponent end fraction left parenthesis a plus b right parenthesis equals fraction numerator 1 over denominator left parenthesis a plus b right parenthesis to the power of 3 end exponent end fraction
    Trick : Put A equals 9 0 to the power of o end exponent, then
    fraction numerator sin to the power of 4 end exponent invisible function application A over denominator a end fraction plus fraction numerator cos to the power of 4 end exponent invisible function application A over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fractionÞfraction numerator 1 over denominator a end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow b equals 0
    therefore fraction numerator sin to the power of 8 end exponent invisible function application A over denominator a to the power of 3 end exponent end fraction plus fraction numerator cos to the power of 8 end exponent invisible function application A over denominator b to the power of 3 end exponent end fraction equals fraction numerator 1 over denominator a to the power of 3 end exponent end fractionwhich is given by option (a)
    Note : Students can check this question for other values of A also.

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