How do you get from this expression to the other? Briefly, I am following the solution to a problem, which I understand up to the point where this expression is reached: 
$$u_x = \frac{vy + \sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)}}{1 + y^2}$$
The solution then states that this is equivalent to: 
$$u_x = \frac{c^2 - v^2}{\sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)} - vy}$$
My question is, how do you get from one to the other? It is possibly obvious, but I just can't see it and it's driving me insane.  

\begin{align}
u_x =& \frac{c^2 - v^2}{\sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)} - vy}\\\\
 =& \frac{\left(\sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)} + vy\right)(c^2 - v^2)}{\left(\sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)} - vy\right)\left(\sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)} + vy\right)}\\\\
=&\frac{\left(\sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)} + vy\right)(c^2 - v^2)}{(c^2 - v^2)(1 + y^2) }\\\\
=& \frac{vy + \sqrt{(vy)^2 + (c^2 - v^2)(1 + y^2)}}{1 + y^2}
\end{align}
