\(\sin x \cdot \cos \varepsilon + \left({\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3 - \sin x\right)\)
- Started with
\[\sin \left(x + \varepsilon\right) - \sin x\]
16.9
- Using strategy
rm 16.9
- Applied sin-sum to get
\[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
6.4
- Applied associate--l+ to get
\[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
6.4
- Using strategy
rm 6.4
- Applied add-cube-cbrt to get
\[\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{red}{\sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3} - \sin x\right)\]
6.7
- Applied add-cube-cbrt to get
\[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\sqrt[3]{\cos x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 - \sin x\right)\]
6.8
- Applied cube-unprod to get
\[\sin x \cdot \cos \varepsilon + \left(\color{red}{{\left(\sqrt[3]{\cos x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3} - \sin x\right)\]
6.8
