Initial program 37.6
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.6
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum22.2
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
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 add-log-exp0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\log \left(e^{\cos \varepsilon \cdot \sin x - \sin x}\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \log \left(e^{\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}}\right)\]
Applied distribute-rgt-out--0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \log \left(e^{\color{blue}{\sin x \cdot \left(\cos \varepsilon - 1\right)}}\right)\]
Applied exp-prod0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \log \color{blue}{\left({\left(e^{\sin x}\right)}^{\left(\cos \varepsilon - 1\right)}\right)}\]
Applied log-pow0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \log \left(e^{\sin x}\right)}\]
- Using strategy
rm Applied flip--0.6
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} \cdot \log \left(e^{\sin x}\right)\]
Applied associate-*l/0.6
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{\left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right) \cdot \log \left(e^{\sin x}\right)}{\cos \varepsilon + 1}}\]
Simplified0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \frac{\color{blue}{\left(-\sin \varepsilon\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}}{\cos \varepsilon + 1}\]
Final simplification0.4
\[\leadsto \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(-\sin \varepsilon\right)}{1 + \cos \varepsilon} + \cos x \cdot \sin \varepsilon\]
