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Prove that (1/(1+x^(b-a)+x^(c-a) ))+(1/(1+x^(b-a)+x^(c-a) ))+(1/(1+x^(b-c)+x^(a-c) ))=1 

Asked by k.menezes432 4th April 2017, 9:22 AM
Answered by Expert
Answer:

Hi Kevin,

Check your Questions. The terms should be as follows. Check the second set of terms.

begin mathsize 16px style Consider comma space LHS
equals fraction numerator 1 over denominator 1 plus straight x to the power of left parenthesis straight b minus straight a right parenthesis end exponent plus straight x to the power of left parenthesis straight c minus straight a right parenthesis end exponent end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of bold left parenthesis bold a bold minus bold b bold right parenthesis end exponent plus straight x to the power of bold left parenthesis bold c bold minus bold b bold right parenthesis end exponent end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of left parenthesis straight b minus straight c right parenthesis end exponent plus straight x to the power of left parenthesis straight a minus straight c right parenthesis end exponent end fraction
equals fraction numerator 1 over denominator 1 plus straight x to the power of straight b straight x to the power of negative straight a end exponent plus straight x to the power of straight c straight x to the power of negative straight a end exponent end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of straight a straight x to the power of negative straight b end exponent plus straight x to the power of straight c straight x to the power of negative straight b end exponent end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of straight b straight x to the power of negative straight c end exponent plus straight x to the power of straight a straight x to the power of negative straight c end exponent end fraction
equals fraction numerator 1 over denominator 1 plus straight x to the power of negative straight a end exponent left parenthesis straight x to the power of straight b plus straight x to the power of straight c right parenthesis end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of negative straight b end exponent left parenthesis straight x to the power of straight a plus straight x to the power of straight c right parenthesis end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of negative straight c end exponent left parenthesis straight x to the power of straight b plus straight x to the power of straight a right parenthesis end fraction
equals fraction numerator 1 over denominator 1 plus begin display style fraction numerator straight x to the power of straight b plus straight x to the power of straight c over denominator straight x to the power of straight a end fraction end style end fraction plus fraction numerator 1 over denominator 1 plus begin display style fraction numerator straight x to the power of straight a plus straight x to the power of straight c over denominator straight x to the power of straight b end fraction end style end fraction plus fraction numerator 1 over denominator 1 plus begin display style fraction numerator straight x to the power of straight b plus straight x to the power of straight a over denominator straight x to the power of straight c end fraction end style end fraction
equals fraction numerator 1 over denominator fraction numerator straight x to the power of straight a plus straight x to the power of straight b plus straight x to the power of straight c over denominator straight x to the power of straight a end fraction end fraction plus fraction numerator 1 over denominator begin display style fraction numerator straight x to the power of straight b plus straight x to the power of straight a plus straight x to the power of straight c over denominator straight x to the power of straight b end fraction end style end fraction plus fraction numerator 1 over denominator begin display style fraction numerator straight x to the power of straight c plus straight x to the power of straight b plus straight x to the power of straight a over denominator straight x to the power of straight c end fraction end style end fraction
equals fraction numerator straight x to the power of straight a over denominator straight x to the power of straight a plus straight x to the power of straight b plus straight x to the power of straight c end fraction plus fraction numerator straight x to the power of straight b over denominator begin display style straight x to the power of straight a plus straight x to the power of straight b plus straight x to the power of straight c end style end fraction plus fraction numerator straight x to the power of straight c over denominator begin display style straight x to the power of straight a plus straight x to the power of straight b plus straight x to the power of straight c end style end fraction
equals fraction numerator straight x to the power of straight a plus straight x to the power of straight b plus straight x to the power of straight c over denominator straight x to the power of straight a plus straight x to the power of straight b plus straight x to the power of straight c end fraction
equals 1 equals RHS
Hence space proved. end style

Answered by Expert 4th April 2017, 12:14 PM
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