Question

Okey please help me out​

Answer 1

Rewrite the second expression as

`\frac1{1+x^m + x^(-n)} = (x^(\ell+n))/(x^(\ell+n) + x^(\ell+m+n)+x^\ell) = (x^(-m))/(x^(-m) + 1 + x^\ell)`

and the third expression as

`\frac1{1 + x^n + x^(-\ell)} = (x^\ell)/(x^\ell + x^(\ell+n) + 1) = (x^\ell)/(x^\ell + x^(-m) + 1)`

so the fractions all have the same denominator.

Then combining the fractions gives the desired result,

`\frac1{1+x^\ell+x^(-m)} + \frac1{1+x^m+x^(-n)} + \frac1{1+x^n+x^(-\ell)} = (1+ x^\ell + x^(-m))/(1+x^\ell+x^(-m)) = \boxed{1}`