<s>[INST] 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!
 [/INST] \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*}
</s>