In a question I was given, in order to solve it, I needed to simplify $\displaystyle \sin\left(\frac{3}{2}x\right)\cos\left(\frac{3}{2} x\right)$ to $\displaystyle \frac{1}{2}\sin 3x$. This is "obvious" related to the identity $\displaystyle 2\sin x \cos x = \sin 2x$.?

So my question is, what's the rule (if any) to simplify these forms of expressions?

Lets use the example of $\displaystyle \sin\left(\frac{3}{\sqrt{2}}x\right)\cos\left(\fra c{3}{\sqrt{2}}x\right)$ or $\displaystyle \sin\left(\frac{4}{5}x\right)\cos\left(\frac{4}{5} x\right)$, what do those simplify to?