Simplifying the following function

[CoinToss]

Homework Statement

I have stumbled upon the following equation on wikipedia
http://upload.wikimedia.org/math/b/8/d/b8de8bfcdf40f1793e7b72755161d61f.png

To clarify, I am not sure how to simplify the equation of the form

x^2/(1-x) = 1/(x^2 - x^-2)

as demonstrated in the second line of the equation.

The Attempt at a Solution

I temporarily reciprocated the equation to become

(1-x)/ x^2 = (x^2 - x^-2)

But,

1/x² - x/x² ≠ (x² - x^-2) ...?

Something is terribly wrong here. Please advice.

 

[Dick]

Homework Statement

I have stumbled upon the following equation on wikipedia
http://upload.wikimedia.org/math/b/8/d/b8de8bfcdf40f1793e7b72755161d61f.png

To clarify, I am not sure how to simplify the equation of the form

x^2/(1-x) = 1/(x^2 - x^-2)

as demonstrated in the second line of the equation.

The Attempt at a Solution

I temporarily reciprocated the equation to become

(1-x)/ x^2 = (x^2 - x^-2)

But,

1/x² - x/x² ≠ (x² - x^-2) ...?

Something is terribly wrong here. Please advice.

I think you are misreading where the exponents of 2 are. The form you want to show is x/(1-x^2)=1/(1/x-x).