Initial program 37.5
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.5
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum21.9
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
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 flip3--0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\left(\color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}\]
Applied associate-*l/0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\color{blue}{\frac{\left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right) \cdot \left(\cos \varepsilon - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} \cdot \left(\cos \varepsilon - 1\right)}\]
Applied associate-*l/0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{\color{blue}{\frac{\left(\left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}}\]
Applied cbrt-div0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{\sqrt[3]{\left(\left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}}{\sqrt[3]{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}}\]
Simplified0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}}}{\sqrt[3]{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\]
Final simplification0.5
\[\leadsto \cos x \cdot \sin \varepsilon + \sin x \cdot \frac{\sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}}{\sqrt[3]{\left(\cos \varepsilon + 1\right) + \cos \varepsilon \cdot \cos \varepsilon}}\]
