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If (cosx)^6 + (sinx)^6 = a1 + a2 cos4x and a1*a2=m/n(where m/n is rational) is in its lowest form, then (m+n) equals?

Asked by ajayagarwal1970 | 22nd Jun, 2014, 08:46: PM

Expert Answer:

$G i v e n space e x p r e s s i o n space i s space cos to the power of 6 x plus sin to the power of 6 x space cos to the power of 6 x plus sin to the power of 6 x equals open parentheses cos squared x close parentheses cubed plus open parentheses sin squared x close parentheses cubed equals open parentheses cos squared x plus sin squared x close parentheses open parentheses cos to the power of 4 x minus cos squared x sin squared x plus sin to the power of 4 x close parentheses equals open parentheses cos to the power of 4 x minus cos squared x sin squared x plus sin to the power of 4 x close parentheses equals open parentheses cos squared x minus sin squared x close parentheses squared plus cos squared x sin squared x equals cos squared 2 x plus fraction numerator sin squared 2 x over denominator 2 end fraction equals cos squared 2 x plus fraction numerator 1 minus cos squared 2 x over denominator 2 end fraction equals fraction numerator 2 cos squared 2 x plus 1 minus cos squared 2 x over denominator 2 end fraction equals fraction numerator 1 plus cos squared 2 x over denominator 2 end fraction equals fraction numerator 1 plus begin display style fraction numerator 1 plus cos 4 x over denominator 2 end fraction end style over denominator 2 end fraction equals fraction numerator 2 plus 1 plus cos 4 x over denominator 2 cross times 2 end fraction equals fraction numerator 3 plus cos 4 x over denominator 4 end fraction equals a subscript 1 plus a subscript 2 cos 4 x rightwards double arrow a subscript 1 equals 3 over 4 space a n d space a subscript 2 equals 1 fourth G i v e n space t h a t space a subscript 1 cross times a subscript 2 equals m over n rightwards double arrow space a subscript 1 cross times a subscript 2 equals 3 over 4 cross times 1 fourth equals 3 over 16 equals m over n rightwards double arrow m plus n equals 16 plus 3 equals 19$

Answered by Vimala Ramamurthy | 23rd Jun, 2014, 03:39: PM

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