I am trying to simplify this expression by as usual the expansion way,
$$\biggl(\frac{ 1+x^2}{1-x^2}\biggr)^2 = \frac{1}{1-y^2}$$
After some steps I am getting:
$$4x^2 - y^2 - 2x^2y^2 - x^4y^2 = 0$$
The answer suggested in my module is $x^2y = 2x - y$
For the answer to be correct I think what I should get is
$$4x^2 - y^2 - 4xy - x^4y^2 = 0$$
What exactly I am doing wrong ? I tried to find an error in my solution, but unable to spot any(yet).
EDIT: For reference I am adding the other options mentioned the question (and now the question too):
if $4\biggl[\frac{x^2}{1} + \frac{x^{6}}{3}+ \frac{x^{10}}{5} + \cdots \biggr] = y^2 + \frac{y^4}{2} + \frac{y^6}{3} + \cdots $, then
$$x^2y = 2x+y \text{ or } x = 2y^2 - 1 \text{ or } x^2y = 2x + y^2$$