how can I prove this identity How can I prove this identity 

$$\cos (3x/2)-\cos(x/2)=-2 \sin (x/2) \sin(x)$$

I try this but does not work with me $$\cos(3x/2) = \cos(x + x/2) = \cos (x) \cos (x/2) - \sin (x) \sin (x/2)$$ 
I do not know what to do after this step.

We have $$\cos(A+B) -\cos(A+B) = (\cos A\cos B -\sin A\sin B)-(\cos A\cos B +\sin A\sin B) = -2\sin A\sin B. $$ Let $A+B = C$ and $A-B = D$. Our equation then becomes $\cos C -\cos D = -2\sin (\frac{C+D}{2})\sin (\frac{C-D}{2})$. With $C=\frac{3x}{2}$ and $D=\frac{x}{2}$, we have $$ \cos(\frac{3x}{2})-\cos(\frac{x}{2}) = -2\sin (x) \sin (\frac{x}{2})$$
