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`

Answer 1

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.