The formula for the sum of angles in cosine is:
[tex]\cos (a+b)=\cos (a)\cdot\cos (b)-\sin (a)\cdot\sin (b)[/tex]
So in this case the error is adding instead of substracting.
Let's correct it:
[tex]\cos (x+\frac{3\pi}{2})=\cos (x)\cos (\frac{3\pi}{2})-\sin (x)\sin (\frac{3\pi}{2})[/tex]
[tex]\cos (x+\frac{3\pi}{2})=\cos (x)\cdot0-\sin (x)\cdot-1[/tex]
[tex]\cos (x+\frac{3\pi}{2})=\sin (x)[/tex]
