1. The problem statement, all variables and given/known data
Verify Picard's Theorem for cos(1/z) at z = 0

2. Relevant equations
The theorem:

3. The attempt at a solution
I have a solution that I don't understand (from a manual). What the manual does is break out z to get
[tex]z = \frac{1}{log(c\pm \sqrt{c^2-1}}[/tex] for an arbitrary complex constant c. Getting there is trivial enough but I don't get the conclusion.

I understand that you can make z arbitrarily small by choosing a k high enough in the logarithm, but intuitively I'd think cos(z) wouldn't leave [-1, 1] no matter how fast it jumped between them. Intuition be damned, I can't see the reasoning at all. Because I can get a very large input in the cosine function I know the function can output any complex number? How?

Guessing it's something rather silly I'm missing here but there you go.
Any help would be appreciated, cheers.

I think you're missing the fact that [itex] \cos{w}[/itex] does not necessarily sit in the interval [itex] [-1,1] [/itex] when [itex] w [/itex] is complex. For example, if [itex] w [/itex] is pure imaginary, then [itex] \cos{w} = \cosh{|w|} [/itex].

So the point is that you can use the unboundedness from a large imaginary part of [itex] w =1/z[/itex] plus the phase degree of freedom from the real part of [itex] w[/itex] to get more or less any complex number you want.

First, better define what you mean by Picard's Theorem. He has two, the little one and the big one.

Little: Every entire function that's not a polynomial has an essential singularity at infinity.

Big: A function with an essential singularity achieves every value, with at most one exception, infinitely often in any neighborhood of the singularity.

I think you want the big one.

First do a Wikipedia on Casorati-Weierstrass, study the example for [itex]e^{1/z}[/itex], then apply that example to your problem.