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: 
It is a common trick in mathematics to add and subtract something, e.g. 
$$x\longrightarrow (x-y)+y.$$
The idea being that the quantity is the same, but it gives you more possible terms to manipulate and (hopefully) simplify. In particular this can help with fractions like yours, 
$$\frac{x}{x+y}\longrightarrow\frac{(x+y)-y}{x+y}=\frac{x+y}{x+y}-\frac{y}{x+y}=1-\frac{y}{x+y}.$$
While for fractions it is usually easy to see what $y$ is necessary to get something that is easily divisible by the denominator, in general the key is knowing how to pick the right $y$... but I don't think there's anything other than intuition (and practice) that can help with that.
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?