\(\frac{(\left(-\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x \cdot \cos \varepsilon\right))_* - \cos x}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\cos x\right))_* - \left(-\sin x \cdot \sin \varepsilon\right)}{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\cos x\right))_* - \left(-\sin x \cdot \sin \varepsilon\right)}}\)
- Started with
\[\cos \left(x + \varepsilon\right) - \cos x\]
39.8
- Using strategy
rm 39.8
- Applied cos-sum to get
\[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
24.6
- Using strategy
rm 24.6
- Applied sub-neg to get
\[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
24.6
- Applied associate--l+ to get
\[\color{red}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}\]
24.6
- Using strategy
rm 24.6
- Applied flip-+ to get
\[\color{red}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{\cos x \cdot \cos \varepsilon - \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}}\]
24.8
- Applied simplify to get
\[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon - \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin x \cdot \left(-\sin \varepsilon\right)}}\]
24.9
- Applied taylor to get
\[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin x \cdot \left(-\sin \varepsilon\right)} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin x \cdot \left(-\sin \varepsilon\right)}\]
24.9
- Taylor expanded around 0 to get
\[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{\color{red}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_*} - \sin x \cdot \left(-\sin \varepsilon\right)} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_*} - \sin x \cdot \left(-\sin \varepsilon\right)}\]
24.9
- Applied simplify to get
\[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin x \cdot \left(-\sin \varepsilon\right)} \leadsto \frac{\sin \varepsilon \cdot \left(-\sin x\right) - \left(\cos x - \cos \varepsilon \cdot \cos x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \left(-\sin x\right)} \cdot \left(\cos \varepsilon \cdot \cos x - \left(\sin \varepsilon \cdot \left(-\sin x\right) - \cos x\right)\right)\]
6.6
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\sin \varepsilon \cdot \left(-\sin x\right) - \left(\cos x - \cos \varepsilon \cdot \cos x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \left(-\sin x\right)} \cdot \left(\cos \varepsilon \cdot \cos x - \left(\sin \varepsilon \cdot \left(-\sin x\right) - \cos x\right)\right)} \leadsto \color{blue}{\frac{(\left(-\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x \cdot \cos \varepsilon\right))_* - \cos x}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\cos x\right))_* - \left(-\sin x \cdot \sin \varepsilon\right)}{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\cos x\right))_* - \left(-\sin x \cdot \sin \varepsilon\right)}}}\]
24.6
