Is there a difference between $y=\frac{\sqrt{1-x}}{\sqrt{1+x}}$ and $y=\sqrt{\frac{1-x}{1+x}}$ Is there a difference between $$y=\frac{\sqrt{1-x}}{\sqrt{1+x}}$$ and $$y=\sqrt{\frac{1-x}{1+x}}$$
If there is a difference, why when I give the square for both equations, they will be equal.
Yes. There are differences. In the region where both of $x-1$ and $x+1$ are negative, the upper one will be wrong and lower one is correct. This region is $(-\infty, -1).$

For complex numbers there are no difference because any number is acceptable under the sqrt (positive or negative)
In general for real numbers, $ sqrt(\dfrac{a}{b}) = \dfrac{sqrt(|a|)}{sqrt(
|b|)} $ for $ (a \wedge b) \epsilon \mathbb{R}$