How does one come up with a Continued Fraction? All over the place on Wikipedia, I see a bunch of identities related to continued fractions, like
$$\arctan x=\cfrac{x}{1+\cfrac{x^2}{3+\cfrac{4x^2}{5+\cfrac{9x^2}{7+...}}}}$$
or
$$\pi=\cfrac{4}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+...}}}}$$
How does one verify these? I haven't even managed to prove a single one of them yet.
Furthermore, aside from proving them, how does one derive them? That is, how does one come up with something like this?
I know how to evaluate simpler continued fractions, like
$$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+...}}}$$
But I don't know how to evaluate anything where the terms on the inside follow some other sort of sequence.
Can anyone provide a proof (or, preferably, a derivation) of one of the two identities above, or some other continued fraction identity? Can anybody give me some way to go about these types of problems?

Note we have Euler's continued fraction formula, which states:
$$a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots +a_{0}a_{1}a_{2}\cdots a_{n}\\=\cfrac  {a_{0}}{1-{\cfrac  {a_{1}}{1+a_{1}-{\cfrac  {a_{2}}{1+a_{2}-{\cfrac  {\ddots }{\ddots {\cfrac  {a_{{n-1}}}{1+a_{{n-1}}-{\cfrac  {a_{n}}{1+a_{n}}}}}}}}}}}}$$
This may easily be proven by induction and lends itself for easy conversion of a series into a fraction.  For example, we know that:
$$\arctan(x)=\sum_{n=0}^\infty\frac{(-2)^nx^{2n+1}}{2n+1}$$
Set $a_0=x$ and inductively,
$$a_{n+1}=\frac{(-1)^{n+1}x^{2n+3}}{2n+3}\prod_{k=0}^n(a_k)^{-1}=-\frac{2n+1}{2n+3}x^2$$
And then after some algebra, you'll be able to derive
$$\arctan(x)=\cfrac{x}{1+\cfrac{x^2}{3+\cfrac{4x^2}{5+\cfrac{9x^2}{7+\ddots}}}}$$
Plug $x=1$ in and do some more algebra and you'll end up with the formula for $\pi$.
