much simpler. all anybody needs to understand is that when you take say RMS, you are taking the Mean of the Square, so that means the Square is what is important, and you have to work in Squares. the Root is just to convert back to the units you started with after your work in Squares is done. obviously the Squares here are because Watt=VV/Ohm so were by working in Squares were treating power as the most important quantity and doing our voltage calcuations so that they give the right power calculations, power being proportional to the Square of the voltage.

No, it becomes zero, because cos(2pi) – cos(0) = 0

I thought it very interesting that u(rms) = sqrt ((a(0)^2 + a(1)^2)/2) rather than
simply u(rms) = a(o) + a(1)/sqrt 2.

One might intuitively think the a(0) DC offset might add to the AC rms directly rather than increase the u(rms) value as a 90 degree out of phase component.

For the capacitor, only the sine wave counts. The RMS value of the AC component is raised to the power of two and multiplied with the capacitor ESR (Equivalent series resistance).
For the inductor, take the RMS value of the signal, as calculated in this article and multiply it with the inductor DCR (DC resistance).

However, at different frequencies, the amplitude of the AC component varies, because both the inductor and capacitor present a reactance that depends of the capacitor/inductor value and signal frequency. So you need to calculate the signal amplitude before calculating the RMS value.

Are you asking what is the RMS value of a DC signal? That is the DC signal level.

Which is the answer to pure dc of RMS 0 or 1