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$\frac{1}{1 + a^{(n - m)}} + \frac{1}{1 + a^{(m - n)}} = ?$

A. 0

B. $\frac{1}{2}$

C. 1

D. $a^{m + n}$

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Correct Answer : C.

\frac{1}{1 + a^{(n - m)}} + \frac{1}{1 + a^{(m - n)}}

= \frac{1}{(1 + \frac{a n}{a m})} + \frac{1}{(1 + \frac{a m}{a n})}

= \frac{a m}{(a m + a n)} + \frac{a n}{(a m + a n)}

= \frac{(a m + a n)}{(a m + a n)}

= 1.

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