Question

Questions number 2 prove that

Answer 1

$\begin{aligned}&\text{LHS}\\\\\implies\ \ &\dfrac{1}{1+x^{b-a}+x^{c-a}}+\dfrac{1}{1+x^{a-b}+x^{c-b}}+\dfrac{1}{1+x^{b-c}+x^{a-c}}\\\\\implies\ \ &\dfrac{1}{1+\dfrac{x^b}{x^a}+\dfrac{x^c}{x^a}}+\dfrac{1}{1+\dfrac{x^a}{x^b}+\dfrac{x^c}{x^b}}+\dfrac{1}{1+\dfrac{x^b}{x^c}+\dfrac{x^a}{x^c}}\\\\\implies\ \ &\dfrac{1}{\left(\dfrac{x^a+x^b+x^c}{x^a}\right)}+\dfrac{1}{\left(\dfrac{x^a+x^b+x^c}{x^b}\right)}+\dfrac{1}{\left(\dfrac{x^a+x^b+x^c}{x^c}\right)}\end{aligned}$

$\begin{aligned}\implies\ \ &\dfrac{x^a}{x^a+x^b+x^c}+\dfrac{x^b}{x^a+x^b+x^c}+\dfrac{x^c}{x^a+x^b+x^c}\\\\\implies\ \ &\dfrac{x^a+x^b+x^c}{x^a+x^b+x^c}\\\\\implies\ \ &1\\\\\implies\ \ &\text{RHS}\end{aligned}$