Initial program 37.2
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied sin-sum21.9
\[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
Taylor expanded around inf 21.9
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \cos \varepsilon\right) - \sin x}\]
Simplified0.4
\[\leadsto \color{blue}{\sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon}\]
- Using strategy
rm Applied flip--0.5
\[\leadsto \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]
Simplified0.5
\[\leadsto \sin x \cdot \frac{\color{blue}{\cos \varepsilon \cdot \cos \varepsilon - 1}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
- Using strategy
rm Applied sub-1-cos0.4
\[\leadsto \sin x \cdot \frac{\color{blue}{-\sin \varepsilon \cdot \sin \varepsilon}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
Applied distribute-frac-neg0.4
\[\leadsto \sin x \cdot \color{blue}{\left(-\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon + 1}\right)} + \cos x \cdot \sin \varepsilon\]
Applied distribute-rgt-neg-out0.4
\[\leadsto \color{blue}{\left(-\sin x \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon + 1}\right)} + \cos x \cdot \sin \varepsilon\]
Simplified0.2
\[\leadsto \left(-\color{blue}{\frac{\sin x \cdot \sin \varepsilon}{1} \cdot \tan \left(\frac{\varepsilon}{2}\right)}\right) + \cos x \cdot \sin \varepsilon\]
Final simplification0.2
\[\leadsto \cos x \cdot \sin \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\]
