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Find the value of : $\frac{1}{1+a^{n-m}}+\frac{1}{1+a^{m-n}}$.
Given: $\frac{1}{1+a^{n-m}}+\frac{1}{1+a^{m-n}}$.
To do: To find the value of: $\frac{1}{1+a^{n-m}}+\frac{1}{1+a^{m-n}}$.
Solution:
$\frac{1}{1+a^{m-n}}+\frac{1}{1+a^{n-m}}$
$=\frac{1}{1+\frac{a^m}{a^n}}+\frac{1}{1+\frac{a^n}{a^m}}$
$=\frac{1}{\frac{a^m+a^n}{a^n}}+\frac{1}{\frac{a^m+a^n}{a^m}}$
$=\frac{a^n}{a^m+a^n}+\frac{a^m}{a^m+a^n}$
$=\frac{a^m+a^n}{a^m+a^n}$
$=1$
Thus, $\frac{1}{1+a^{n-m}}+\frac{1}{1+a^{m-n}}=1$.
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