Question

1/(1 + x ^ a + x ^ b) + 1/(1 + x ^ (- a) + x ^ (b - a)) + 1/(1 + x ^ (- b) + x ^ (a - b)) = 1 proof​

Answer 1

Answer:

See explanation

Step-by-step explanation:

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

Using  $\boxed{a^{-m}=\dfrac{1}{a^m} \text{ and } a^{m-n}=\dfrac{a^m}{a^n}}$

     $=\dfrac{1}{1+x^a+x^b}+\dfrac{1}{1+\frac{1}{x^a}+\frac{x^{b}}{x^a}}+\dfrac{1}{1+\frac{1}{x^b}+\frac{x^a}{x^b}}\\\\=\dfrac{1}{1+x^a+x^b}+\dfrac{1}{\frac{x^a+1+x^{b}}{x^a}}+\dfrac{1}{\frac{x^b+1+x^a}{x^b}}\\\\=\dfrac{1}{1+x^a+x^b}+\dfrac{x^a}{1+x^a+x^b}+\dfrac{x^b}{1+x^a+x^b}\\\\=\dfrac{1+x^a+x^b}{1+x^a+x^b}=1=\text{RHS}$

Hence proved