Don't understand working out for phasor of output voltage
The sign of the angle changed because it went from denominator to the numerator. That's what taking the inverse of the imaginary unit does.

You can see that more clearly if you just invert the complex number from your first equation:



If Vs=10 as you say, then that derivation is incorrect between the 2nd and 3rd line though, becuase the 2nd line does not equal the 3rd, instead you get 9.956 as the radius/magnitude:



So I suspect Vs is actually \$\frac{10}{\sqrt{2}}\$ or there was some mistake (either on his part in deriving/explaining or when you transcribed it).