Prove that:
$\dfrac{1}{1 + x^{b - a} + x^{c - a}} + \dfrac{1}{1 + x^{a - b} + x^{c - b}} + \dfrac{1}{1 + x^{b - c} + x^{a - c}} = 1$

Given,

$\dfrac{1}{1 + x^{b - a} + x^{c - a}} + \dfrac{1}{1 + x^{a - b} + x^{c - b}} + \dfrac{1}{1 + x^{b - c} + x^{a - c}} = 1$

Solving L.H.S :

Multiplying numerator and denominator of first term by x^a, second term by x^b, third term by x^c we get,

$\Rightarrow \dfrac{1 \times x^a}{(1 + x^{b - a} + x^{c - a})\times x^a} + \dfrac{1 \times x^b}{(1 + x^{a - b} + x^{c - b})\times x^b} + \dfrac{1 \times x^c}{(1 + x^{b - c} + x^{a - c})\times x^c} \\[1em] \Rightarrow \dfrac{1 \times x^a}{(1 \times x^a + x^{b - a} \times x^a + x^{c - a} \times x^a)} + \dfrac{1 \times x^b}{(1 \times x^b + x^{a - b} \times x^b + x^{c - b} \times x^b)} + \dfrac{1 \times x^c}{(1 \times x^c + x^{b - c} \times x^c + x^{a - c} \times x^c)} \\[1em] \Rightarrow \dfrac{x^a}{(x^a + x^{b - a + a} + x^{c - a + a})} + \dfrac{x^b}{(x^b + x^{a - b + b} + x^{c - b + b})} + \dfrac{x^c}{(x^c + x^{b - c + c} + x^{a - c + c})} \\[1em] \Rightarrow \dfrac{x^a}{(x^a + x^b + x^c)} + \dfrac{x^b}{(x^b + x^a + x^c)} + \dfrac{x^c}{(x^c + x^b + x^a)} \\[1em] \Rightarrow \dfrac{x^a + x^b + x^c}{(x^a + x^b + x^c)} \\[1em] \Rightarrow 1.$

Hence proved, $\dfrac{1}{1 + x^{b - a} + x^{c - a}} + \dfrac{1}{1 + x^{a - b} + x^{c - b}} + \dfrac{1}{1 + x^{b - c} + x^{a - c}} = 1$.