Is there a spectrum difference in amplitude modulation with a sine or cosine?

[DragonPetter]

If I modulate a pulse X(t) with a.) a sine wave or b.) a cosine wave, I have the frequency spectrum expressions
$$a.) \frac{1}{2j}[X(f-f_0)-X(f+f_0)]$$
$$b.) \frac{1}{2}[X(f-f_0)+X(f+f_0)]$$

When I plot these for a pulse, I see a difference in the magnitude spectrum, but I should not expect to see this since this is simply a phase shift of $$\pi/2$$

I would think if I multiplied b.) by $$e^{-j\pi/2}$$, I would get what I see in a.) but this is not the case.

Can anyone help me with what I'm missing?

 

[DragonPetter]

Ok, I'm guessing the phase shift e^-jpi/2 does not extract out so nicely as I thought for the a.) case when the sine Fourier transform is taken. The two components X(f-fo) and X(f+fo) will have opposite phases unlike in b.) so I cannot simply multiply b.) by a phase shift to get to a.)