Simplify $\dfrac{\sqrt{m+x}+\sqrt{m-x}}{\sqrt{m+x}-\sqrt{m-x}}$

Simplify $$\dfrac{\sqrt{m+x}+\sqrt{m-x}}{\sqrt{m+x}-\sqrt{m-x}}$$ if $x=\dfrac{2mn}{n^2+1}$ and $m>0,n>1$.

The solution of the authors starts as follows: $$\dfrac{\left(\sqrt{m+x}+\sqrt{m-x}\right)^2}{(\sqrt{m+x})^2-(\sqrt{m-x})^2}=\dfrac{m+x+2\sqrt{m^2-x^2}+m-x}{m+x-m+x}=\dfrac{m+\sqrt{m^2-x^2}}{x}=...$$ I don't get the idea behind this. What exactly have they done with the given expression? Thank you!

Solution 1:

\begin{align*} \bigg(\dfrac{\sqrt{m+x}+\sqrt{m-x}}{\sqrt{m+x}-\sqrt{m-x}}\bigg) \bigg(\dfrac{\sqrt{m+x}+\sqrt{m-x}}{\sqrt{m+x}+\sqrt{m-x}}\bigg) \\ \\ =\dfrac{\big(\sqrt{m+x}+\sqrt{m-x}\big)^2} {(m+x)-(m-x)}\\ \\ =\dfrac{2 \sqrt{m - x} \sqrt{m + x} + 2 m} {2x}\\ \\ =\dfrac{\sqrt{m^2 - x^2} + m} {x} ,\space x=\dfrac{2mn}{n^2+1},\space m>0,\space n>1 \end{align*}