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Question
Prove that:
`1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))=1`
Solution
Consider the left hand side:
`1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))`
`=1/(1+x^b/x^a+x^c/x^a)+1/(1+x^a/x^b+x^c/x^b)+1/(1+x^b/x^c+x^a/x^c)`
`=1/((x^a+x^b+x^c)/x^a)+1/((x^b+x^a+x^c)/x^b)+1/((x^c+x^b+x^a)/x^c)`
`=x^a/(x^a+x^b+x^c)+x^b/(x^b+x^a+x^c)+x^c/(x^c+x^b+x^a)`
`=(x^a+x^b+x^c)/(x^a+x^b+x^c)`
= 1
Therefore left hand side is equal to the right hand side. Hence proved.
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