Suppose we use a rope, and pull start the 750 watt motor to avoid the inrush start currents and the inverter could now support it.
The VA at the motor will be about 1150VA, the watts at the motor will be 750 watts..... assuming 100% efficiency in the inverter, will the DC VxA=750 watts?
It's not DC VxA. It's in phase (with the voltage) current versus 90 degrees out-of-phase current.
(The resistive/reactive analysis is for pure sine waves. For other waveforms you can sometimes treat them as a sum of harmonics and sometimes just have to throw up your hands and integrate everything across the whole waveform. B-( So I'll stick with the simple sine-wave interpretation from here on - as power engineering guys tend to do because it usually dominates all the other stuff.)
The in-phase portion of the current represents the resistive load (or source, if it's negative, i.e. 180 degrees out-of-phase). The 90 degrees out-of-phase ("reactive") component of the current represents the current into the inductive or capacitive part of the load. This occurs a quarter-cycle ahead of the voltage ("leading") if the load is capacitive, a quarter-cycle behind ("lagging") if the load is inductive. Because they're 90 degrees out-of-phase their product integrates out to zero over a full cycle and for a lot of things you can treat the restive and reactive current as separate. "Real power" - the product of the in-phase RMS current with the RMS voltage - represents the resistive load - power delivered from the supply to the load on every cycle. It's measured in Watts. "Imaginary power" or "reactive power" - the product of the reactive RMS current with the RMS voltage - represents energy swapped back-and-forth between the supply and the load every half-cycle. It's measured in VARs (Volt Amps Reactive). Capacitive and inductive reactive currents are opposite in sign, so which is "gerating" and which is "consuming" VARs is arbitrary. Because most of the parasitic loads on a power grid are inductive, then convention is to chose the sign so that that inductors "consume" VARs and capacitors "generate" them. (The power company has to either hang extra capacitors on the line to "generate" VARs, and any they don't generate with capacitors are generated by out-of-phase currents in the actual generators.)
(The "real" versus "imaginary" nomenclature comes from mathematical treatment of the magnitude and phase of a current as a complex number. The "imaginary" part of the current is just as real in the ordinary sense as the "real" part, which can cause confusion when talking about it. So while academics use the confusing nomenclature and embarrass new students, people working in the field tend to stick with "reactive" when referring to the 90-degree currents. When talking about a current-voltage product they'll still use "real power" for the in-phase, producer-to-consumer flow and sometimes say "imaginary power" rather than "reactive power" when talking about the out-of-phase, ping-pong flow.)
Losses in wiring and transformer windings are resistive losses - proportional to the square of the current and unrelated to the voltage between the conductor and anything else. (That's why transformer "power" ratings are in VA rather than Watts.) The "they're separate" approximation doesn't hold when you want to compute the overall current in the wiring from the combination of real and reactive current. Instead you have to add the resistive and reactive current using vector addition. Think of the resistive current as the length of an arrow from left to right, reactive current as the length of an arrow from that one's tip and pointing up for leading (capacitive), down for lagging (inductive). Then draw an arrow from the base of the first to the tip of the second. Right triangle: Compute it's length with the square-root of sum of squares rule.
Power factor is the ratio of the amount of the in-phase current to the overall load current (that vector sum of the resistive and reactive current). It represents the fraction of the power that you'd expect from looking at the current to the amount you actually get delivered, rather than playing ping-pong with your supply.
Back to your motor: Yes. The real power will be 750 watts. The VA will be 1150. With a little trig we can see that the reactive power (which is inductive) will be sqrt ( 1150^2 - 750^2 ) =~ 872 VAR.
Now we place a run cap on the motor (in series with the start field), and our AC VA comes down to 750W (PF=1), our motor watts is about the same .. 750 watts (forget cap losses... small) does our DC VxA change at all?... I expect it to remain about the same (unless there was a lot of line loss in the AC side).
(Again changing "DC VxA" to "V x in-phase A", "real power", or "Watts"...) Yes, the real power into your run coil is still 750W. However you've just added another coil...
The reactive current in that coil is phase shifted more leading by the capacitor, so it will cancel some of the lagging reactive current of the run coil. But the resistive losses in the coil - from coil resistance, eddy-current losses in the core, and rotor load, along with the resistive losses in the capacitor, still show up as an additional resistive load current on the input. (Rotor load showing up on the start coil will have been lifted from the run coil's load but the rest will be new loads.)
So adding an always-on run cap to the start coil does help significantly (in addition to reducing vibration in the driven shaft). But hanging a capacitor directly across the run winding, as Flux described, will do a more pure job of canceling that coil's reactive current while only adding the capacitor's internal resistive losses to the main load.