Initial program 36.8
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification36.8
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum21.6
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
Applied associate--l+0.4
\[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}\right)\]
Applied prod-diff0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left((\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_* + (\left(-\sin x\right) \cdot 1 + \left(\sin x \cdot 1\right))_*\right)}\]
Applied associate-+r+0.4
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + (\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*\right) + (\left(-\sin x\right) \cdot 1 + \left(\sin x \cdot 1\right))_*}\]
Simplified0.4
\[\leadsto \left(\sin \varepsilon \cdot \cos x + (\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*\right) + \color{blue}{(-1 \cdot \left(\sin x\right) + \left(\sin x\right))_*}\]
- Using strategy
rm Applied add-log-exp0.5
\[\leadsto \left(\sin \varepsilon \cdot \cos x + \color{blue}{\log \left(e^{(\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*}\right)}\right) + (-1 \cdot \left(\sin x\right) + \left(\sin x\right))_*\]
Final simplification0.5
\[\leadsto (-1 \cdot \left(\sin x\right) + \left(\sin x\right))_* + \left(\log \left(e^{(\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x\right))_*}\right) + \cos x \cdot \sin \varepsilon\right)\]
