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 xx x^-a + x^c xx x^-a) + 1/(1 + x^a xx x^-b + x^c xx x^-b) + 1/(1 + x^b xx x^-c + x^a xx x^-c)`   ...[∵ a^{m + n} = a^m × aⁿ]

= `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^a + x^b + x^c) + x^c/(x^a + x^b + x^c)`

= `(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.