surprising consequences in SR

bernhard.rothenstein

Consider the Dopper shift formula from two inertial reference frames I and I' in the standard arrangement and intial conditions, in relative motion wih b=V/c
f'=g(V)f(1-bcosw) (1)
f=g(V)f'(1+bcosw'). (2)
I do not discuss the way in which they are derived! Eliminating f and f' between (1) and (2) we obtain that the angles w and w' are related by
cosw=(cosw'+b)/(1+bcosw') (3)
cosw'=(cosw-b)/(1-bcosw). (4)
We recognise in (3) and (4) the formulas which account for the aberration of light effect.
So far as I know (3) and (4) hold only in the case when a source of light located at t=t'=0 at the point where the origins of the two frames are instantly located the times t=t'=0 being displayed by the clocks C and C' located at the corresponding origins. So, the derivation of (3) and (4) does not involve clock synchronization, involving instead the initialization of the two clocks.As a consequence IMHO the derivation of (1) and (2) does not require clock synchrnization.
As it is shown by
Robert Resnick, "Introduction to Special Relativity," John Willey and Sons. Inc. shows that (3) and (4) can be derived from the addition law of relativistic velocities applied in the case of a light signal.
Ar those facts a simple coincidence?
If not, what is the consequence for the magnitude of f? Take into account the fact that we can transmit light signals at very low frequencies but with c. Consider that the addition law of velocities relate instantaneous velocities, should f be an instantaneous frequency largely discussed by those who work with transmission of information with light signals.
I underline that I do not put under question SR, but I follow Einstein's hint:
Do not stop thinking! I think he would smile if I am wrong.