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-sum22.3
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
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-cbrt-cube0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}}\]
- Using strategy
rm Applied flip--0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}}}\]
Applied flip--0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}}\right) \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}}\]
Applied flip--0.6
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\left(\color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}\right) \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}}\]
Applied frac-times0.6
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\color{blue}{\frac{\left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right) \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)}{\left(\cos \varepsilon + 1\right) \cdot \left(\cos \varepsilon + 1\right)}} \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}}\]
Applied frac-times0.6
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\color{blue}{\frac{\left(\left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right) \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)\right) \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)}{\left(\left(\cos \varepsilon + 1\right) \cdot \left(\cos \varepsilon + 1\right)\right) \cdot \left(\cos \varepsilon + 1\right)}}}\]
Applied cbrt-div0.6
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right) \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)\right) \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)}}{\sqrt[3]{\left(\left(\cos \varepsilon + 1\right) \cdot \left(\cos \varepsilon + 1\right)\right) \cdot \left(\cos \varepsilon + 1\right)}}}\]
Simplified0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \left(-\sin \varepsilon\right)}}{\sqrt[3]{\left(\left(\cos \varepsilon + 1\right) \cdot \left(\cos \varepsilon + 1\right)\right) \cdot \left(\cos \varepsilon + 1\right)}}\]
Final simplification0.4
\[\leadsto \cos x \cdot \sin \varepsilon + \frac{-\sin \varepsilon \cdot \sin \varepsilon}{\sqrt[3]{\left(\cos \varepsilon + 1\right) \cdot \left(\left(\cos \varepsilon + 1\right) \cdot \left(\cos \varepsilon + 1\right)\right)}} \cdot \sin x\]
