Question: For being a guy who dreams of studying medicine one day, it hurts me to ask such a question because it makes me feel like a total idiot. Anyways... 
$$1 - \frac{1}{1+x^2} = \frac{x^2}{1+x^2}$$

I can go from the left part to the right part, but not the other way. I just don't understand what the trick is her. It's not obvious to me. I came a across this rewrite while doing integration by parts. 

Any mental-math tip for me?

Answer: 
Just add zero to the numerator:

$$\frac{x^2}{1+x^2} = \frac{(1 - 1) + x^2}{1+x^2} = \frac{1+x^2}{1+x^2}-\frac{1}{1+x^2} = 1 - \frac{1}{1+x^2}$$
This is just polynomial division: if $A(x)$ and $B(x)$ are polynomials with $\deg(b) \ge 1$, then $A(x)/B(x)$ can always be expressed as $Q(x) + R(x)/B(x)$, with $\deg(R) < \deg(B)$. 

Do you know how to do this?