Beweis Ableitung Sinus/Cosinus
1. Satz
\begin{align*}
\sin(x)' =& \cos(x) \\
\cos(x)' =& -\sin(x)
\end{align*}
2. Lemma
3. Beweis
3.1. Cosinus
\begin{align*}
\cos(\theta)' =& \lim_{\varepsilon \to 0}\left(\frac{\cos(\theta + \varepsilon)-\cos(\theta)}{\varepsilon}\right) \\
=& \lim_{\varepsilon \to 0}\left(\frac{\cos(\theta)\cos(\varepsilon) - \sin(\theta)\sin(\varepsilon) - \cos(\theta)}{\varepsilon}\right) \\
=& -\cos(\theta) \cdot \lim_{\varepsilon \to 0}\left(\frac{1 - \cos(\varepsilon)}{\varepsilon}\right) - \sin(\theta) \cdot \lim_{\varepsilon \to 0}\left(\frac{\sin(\varepsilon)}{\varepsilon}\right) \\
=& -\cos(\theta) \cdot 0 - \sin(\theta) \cdot 1 \\
=& -\sin(\theta)
\end{align*}