Hello, I'm doing an induction problem and in the basis step I have to verify this identity (assuming $\displaystyle \sin{x/2}\neq 0$):

$\displaystyle \sin{x}=\frac{\sin{(2x)}}{4\sin^2{\left(\frac{x}{2 } \right) }}-\frac{2\cos{\left( \frac{3x}{2} \right) }}{2\sin{\left( \frac{x}{2} \right) }}$

Getting a common denominator, I get the RHS to be

$\displaystyle \frac{\sin{(2x)}-4\sin{\left( \frac{x}{2} \right) }\cos{\left( \frac{3x}{2} \right) }}{4\sin^2{\left( \frac{x}{2} \right) }}$

But where do I go from here? I'm afraid I have almost no background in trigonometry, and so don't know the strategy for these types of problems.