$δ,ε$-proof for limit $\frac{\sqrt{2-2\cos(x)}}{x}$. I am trying to prove the following limit.
$$
\lim_{\Delta\theta\rightarrow0+}\frac{\sqrt{2-2\cos\Delta\theta}}{\Delta\theta}=1 $$
I am going to use this result to prove derivatives of sine and cosine, so I am looking for a proof that won't compute this one by using L'Hôpital's rule. Maybe there is a δ,ε-proof or something else?

First, $\displaystyle 2-2\cos x = 4 \sin^2 \Big( \frac x2 \Big),$ and so, for $x >0$,
$$\sqrt{2-2\cos x} = 2\, \Big|\sin \Big( \frac x2 \Big)\Big| = 2 \sin \Big( \frac x2 \Big).$$ Now your limit simply becomes:
$$\lim_{x \to 0^+} \dfrac{\sin \Big( \dfrac x2 \Big)}{\dfrac x2} = \lim_{t \to 0^+} \frac{\sin t}{t}.$$
