TITLE: How to prove $\exp(x)/(\exp(x)+1)^2$ is even?
QUESTION [12 upvotes]: I've spent some time trying to prove that the function:
$$f(x)=\frac{\exp x}{(\exp x+1)^2}$$
is even. 
I tried expanding the different $\exp x$ as power series, but I had a very difficult time trying to track the different indices. Is that the correct way to proceed or is there some other property that I am no taking into account ? 
Greetings.

REPLY [5 votes]: Something that may be more generally useful: instead of proving that $f(x) = f(-x)$ directly, it's sometimes easier to show that $f(x) - f(-x) = 0$. Writing out $f(x)$ and $f(-x)$ and trying to combine terms may suggest a simplification that you wouldn't think of otherwise. In this case, you'd start with
$$\frac{e^x}{(e^x + 1)^2} - \frac{e^{-x}}{(e^{-x} + 1)^2}$$
(I don't think there's any point in hiding the specific steps, since they appear in several other answers.)
You might recognize immediately that multiplying one of these terms by $e^{\pm 2x}$ on the top and bottom converts it into the other term. If not, you can try the classic technique for adding fractions with different denominators:
$$\frac{e^x(e^{-x} + 1)^2}{(e^x + 1)^2(e^{-x} + 1)^2} - \frac{e^{-x}(e^{x} + 1)^2}{(e^{-x} + 1)^2(e^x + 1)^2}$$
and then it should be pretty clear that you should expand the numerators, after which the result is fairly obvious.
The same idea goes for proving that a function is odd, just show that $f(x) + f(-x) = 0$ instead of $f(x) - f(-x)$.