Initial program 36.9
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification36.9
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum21.7
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
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-cbrt-cube0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)}}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \left(\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}\right)}\]
Applied distribute-rgt-out--0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \color{blue}{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}}\]
- Using strategy
rm Applied flip3--0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \left(\sin x \cdot \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\right)}\]
Final simplification0.4
\[\leadsto \cos x \cdot \sin \varepsilon + \sqrt[3]{\left(\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\left(1 + \cos \varepsilon\right) + \cos \varepsilon \cdot \cos \varepsilon}\right) \cdot \left(\left(\sin x \cdot \cos \varepsilon - \sin x\right) \cdot \left(\sin x \cdot \cos \varepsilon - \sin x\right)\right)}\]
