N2O3
Determine the number of moles of oxygen and nitrogen you have:
Atomic weight nitrogen = 14.0067
Atomic weight oxygen = 15.999
Moles nitrogen = 7.37 / 14.0067 = 0.526176758 moles
Moles oxygen = (20 - 7.37) / 15.999 = 12.63 / 15.999
= 0.789424339 moles
So the nitrogen to oxygen ratio is
0.526176758 : 0.789424339
We need a ratio of small integers that's close to the above ratio, so divide by the smallest number in ratio, getting
1 : 1.500302564
The 1.500 isn't close, but it does indicate that multiplying by 2 should be useful, giving
2 : 3.000605127
And the ratio 2:3 looks good. So the empirical formula for the unknown compound is N2O3
