Initial program 37.4
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.4
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum22.1
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
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 distribute-rgt-out--0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(\cos \varepsilon - 1\right)}\]
- Using strategy
rm Applied add-log-exp0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\log \left(e^{\cos \varepsilon - 1}\right)}\]
- Using strategy
rm Applied add-sqr-sqrt0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \log \color{blue}{\left(\sqrt{e^{\cos \varepsilon - 1}} \cdot \sqrt{e^{\cos \varepsilon - 1}}\right)}\]
Applied log-prod0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos \varepsilon - 1}}\right) + \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right)\right)}\]
Final simplification0.5
\[\leadsto \cos x \cdot \sin \varepsilon + \left(\log \left(\sqrt{e^{\cos \varepsilon - 1}}\right) + \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right)\right) \cdot \sin x\]
