Prove that:1 1+x^ a-b +1 1+x^ b-a =1 - Vidyadip

Question

Prove that:$\frac{1}{1+\text{x}^{\text{a}-\text{b}}}+\frac{1}{1+\text{x}^{\text{b}-\text{a}}}=1$

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Answer

$\frac{1}{1+\text{x}^{\text{a}-\text{b}}}+\frac{1}{1+\text{x}^{\text{b}-\text{a}}}=1$Left hand side (LHS) = Right hand side (RHS) Considering LHS,
$=\frac{1}{1+\frac{\text{x}^\text{a}}{\text{x}^\text{b}}}+\frac{1}{1+\frac{\text{x}^\text{b}}{\text{x}^\text{a}}}$
$=\frac{\text{x}^\text{b}}{\text{x}^\text{b}+\text{x}^\text{a}}+\frac{\text{x}^\text{a}}{\text{x}^2+\text{x}^\text{b}}$
$=\frac{\text{x}^\text{b}+\text{x}^\text{a}}{\text{x}^\text{a}+\text{x}\text{b}}$
$=1$
Therefore, LHS = RHS Hence proved

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