Question

1/1+a^m-n + 1/1+a^n-m
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Answer 1

Answer:

$\frac{1}{1 + {a}^{m - n } } + \frac{1}{1 + {a}^{n - m } } \\ \\ = \frac{1 + {a}^{n - m } +1 + {a}^{m - n } }{(1 + {a}^{m - n })(1 + {a}^{n - m})} \\ \\ = \frac{2 + {a}^{m - n } + {a}^{n - m }}{1 + {a}^{n - m } + {a}^{m - n } + {a}^{m - n } \times {a}^{n - m } } \\ \\ = \frac{2 + {a}^{m - n } + {a}^{n - m }}{1 + {a}^{n - m } + {a}^{m - n } + {a}^{m - n + n - m} } \\ \\= \frac{2 + {a}^{m - n } + {a}^{n - m }}{1 + {a}^{n - m } + {a}^{m - n } + {a}^{0} } \\\\ = \frac{2 + {a}^{m - n } + {a}^{n - m }}{1 + {a}^{n - m } + {a}^{m - n } + 1 } \\ \\= \frac{2 + {a}^{m - n } + {a}^{n - m }}{2 + {a}^{m - n } + {a}^{n - m } } \\ \\ = 1 \\ \\Thus\\ \\ \fbox{\therefore \frac{1}{1 + {a}^{m - n } } + \frac{1}{1 + {a}^{n - m } } = 1}$