Simplify $x^2\sin{\left({\frac{x}{2}}\right)}\cos{\left({\frac{x}{2}}\right)}$

Question

In this problem I need to simplify the following expression:

$$x^2\sin{\frac{x}{2}}\cos{\frac{x}{2}}$$

However, I can't think of any trigonometric properties or algebraic manipulations that would simplify it. I know this isn't a proper way to ask a question here, without showing any working, but any ideas or hints would be appreciated :)

Answer 1

Recall that

$$\sin 2\theta =2\sin \theta \cos \theta \implies \sin \theta \cos \theta=\frac12\sin 2\theta $$

therefore with $\theta=\frac x 2$ we have

$$x^2\sin{\frac{x}{2}}\cos{\frac{x}{2}}=\frac12x^2\sin x$$

Answer 2

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

Thus we have $$ x^2\sin{\frac{x}{2}}\cos{\frac{x}{2}}=\frac12x^2\sin x$$

No more simplification is possible.