But I am wondering what is the proof behind why kirchcoffs rules still apply to phasors.

What I mean is once you convert the sine voltage/ current functions into phasors. Why does all the laws still hold with working with phasors?

And why can you add phasors and translate back into a sine wave that would be equivalent to do the arithmetic with trig function identities always.

I hope you don't miss understand this question.

I know how to do the math but I don't fully understand the proof why the phasor kirchcoffs rules hold. I see no link between them.

One is a number (phasor ) that represents trig function (sin wave voltage ac ...etc)

The other is the actually equation for voltage , current etc etc...

Note I am looking for proof.

I can kind of see the translation between phasors to trig function and visa versa ( except it is not a one to one corospondences)

Anyway I will stop rambling on.

Thanks for any help.