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Let $Φ(t) = 1 + a^t$

Show that $1/Φ(t) + 1/Φ(-t) = 1$

I'm not sure where to start on this one. We've just started exponential functions, so I'm going to assume I just subsitute in $1 + a^t$ for $Φ(t)$. I guess I need to then determine what $Φ(-t)$ equals based on the fact that $Φ(t) = 1 + a^t$. Then substitute that information into the equation and it will probably look like a more familiar equation. Am I on the right track?

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    Yes, you're on the right track. Why don't you try doing what you've described and see where it leads you?2011-11-22

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Yes. Simply do the substitution correctly and use the fact that $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ and with some algebra you will be done.

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$ \frac{1}{1+a^{-t}} $ If you multiply both the numerator and the denominator of this fraction by $a^t$, then the denominator is $a^t(1+a^{-t})$, and that bears simplification. After that, it's easy.

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$\dfrac{1}{\Phi(t)} + \dfrac{1}{\Phi(-t)} = \dfrac{1}{1 + a^t} + \dfrac{1}{1 + a^{-t}}$

Can you take it from here?

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    Using the hint that Michael Hardy provided, hopefully you'll have a straight shot to a solution!2011-11-22