I fail to see how does the output is always at the same voltage for both halfs of an AC cycle. What I mean is that let's pretend Vin is 10V. Now if R=Xc, the voltages across both components are the same (5V). But what if for example the frequency of the signal decreases a bit. This will increase Xc and now more voltage will fall across the capacitor. Let's say that now 4V are attained across R and 6V across Xc. Here it gets complicated and I hope you help me with this one:
Does this mean that for one half of the cycle, when the +10V are on the side of R you will have 10-4=6V output, and for the other half when +10V are at the side of the capacitor you will get 10-6=4V output?

 

No. Where did you get that idea?
Due to the voltage output phase shift from the capacitor, the voltages add as vectors not as scalars.
You need to use complex arithmetic to solve for the voltages as shown here.
Thus when R =Xc the peak voltage across both is ≈7V, not 5V (although they will both be 5V when the input peaks at 10V).

Below is an LTspice simulation of the circuit for a frequency at which R=Xc.
V(In,Out) is the voltage across the resistor.
Edit: If you are interested in exploring this further I suggest you download a free copy of LTspice from Linear Technology and have at it with the Draft28.asc file attached.

 

No. Where did you get that idea?
Due to the voltage output phase shift from the capacitor, the voltages add as vectors not as scalars.

I was trying to imagine you saying that and gesturing with your coffee cup - just like Wally.

 

Hmm, turns out I can get into alot of confusion trying to physically explain why the output voltage is the same for both halfs of the ac cycle, although I know the theory of ac circuits, phase shift and such. I choose to go along with this --> If you take your output across the cap you get the voltage across the cap and if you take your output across the resistor you get the voltage across the resistor. And dont know if the next observation of mine is true, but no matter what the voltage at the mid point is, the potential difference between the two ends of the capacitor or the resistor will always end up the same.