When I work out the LHS I get cos(x) as my answer, how do I get to the answer on the RHS?

Question

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

Answer 1

Notice

$$ \cos(2\alpha) = \cos^2 \alpha - \sin^2 \alpha$$

Putting $\alpha = \frac{x}{2} $ gives your result.

Answer 2

Hint

$$\frac{\sin \left(2x\right)}{2\sin \left(x\right)}=\frac {2 \sin(x) \cos(x)}{2 \sin(x)}=\cos(x)$$ Now, apply the half angle formula for $\cos(x]$ and you will get the rhs.

I am sure that you can take from here.