Instead of writing e^(-x), try writing it as 1/(e^x). This should clarify it more.

Leave the denominator alone and just work with the numerator:

(e^x + 1/(e^x))(e^x + 1/(e^x)) - (e^x - 1/(e^x))(e^x - 1/(e^x))

Multiply out and you get:

e^2x +1 +1 + (1/(e^2x)) - [e^2x - 1 - 1 + (1/(e^2x))]

Simplify:

e^2x +2 + (1/(e^2x)) - e^2x +2 - (1/(e^2x))

which simplifies to:

4

So, in the end you have:

4 / (e^x + e^-x)^2

Hope this helps