Question

Simplify :
$\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{a-c}+x^{b-c}}$

Answer 1

$\Rightarrow \frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{a-c}+x^{b-c}}$

$\Rightarrow \frac{1}{1+\frac{x^b}{x^a}+\frac{x^c}{x^a}}+\frac{1}{1+\frac{x^a}{x^b}+\frac{x^c}{x^b}}+\frac{1}{1+\frac{x^a}{x^c}+\frac{x^b}{x^c}}$

$\Rightarrow \frac{x^a}{x^a+x^b+x^c}+\frac{x^b}{x^b+x^a+x^c}+\frac{x^c}{x^c+x^a+x^b}$

$\Rightarrow \frac{1}{x^a+x^b+x^c}(x^a+x^b+x^c)$

$\Rightarrow \frac{x^a+x^b+x^c}{x^a+x^b+x^c}\Rightarrow x^0=1$