Can I convert $2\cos x$ to $\sin \dfrac{x}{2}$?

Question

Can I convert $2\cos x$ to $\sin \dfrac{x}{2}$? Or at least $-2+2 \cos x$ to $\sin $.

Answer 1

$$\left|\sin\frac x2\right|=\sqrt{\frac{1-\cos x}2}.$$

Answer 2

Neither of those is a valid substitution. For the first, observe that, at $x = 0$, we have $$ 2\cos 0 = 2\\ \sin \dfrac{0}{2} = 0. $$ For the second, look at $x = \pi$ and notice that $$ -2 + 2 \cos \pi = -4 \\ \sin \dfrac{\pi}{2} = 1. $$