Find LF component of AM signal

I'm trying to figure out how to find the low frequency component of an AM-signal when I have regained the carrier.

So the signal is: $$U_{AM} = 2[3+0.4cos(2\pi\cdot2\cdot 10^3\cdot t)]\cdot sin(2\pi\cdot8\cdot10^6\cdot t)$$

And the extracted carrier is: $$U_{BB}= 2sin(2\pi\cdot35\cdot10^6\cdot t)$$

To find the LF component $$U_{LF} = U_{AM} \times U_{BB}$$

And this gives: $$3+0.4cos(2\pi\cdot2\cdot10^3\cdot t)-3cos(2\pi\cdot8\cdot10^6\cdot t)-0.4cos(2\pi\cdot2\cdot10^3\cdot t)\cdot cos(2\pi\cdot8\cdot10^6\cdot t)$$

Now I'm supposed to filter this to remove the DC and HF component and remain with the LF component. I know the answer is supposed to be: $$0.4cos(2\pi\cdot2\cdot10^3\cdot t)$$ But I'm not sure what the reasoning behind getting this answer is. Does anyone have a good way of explaining to me, once I have the calculated $$U_{LF}$$ how to reason and filter away the HF and DC component?