\(\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3} - \left({\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3 + \cos x\right)\)
- Started with
\[\cos \left(x + \varepsilon\right) - \cos x\]
18.6
- Using strategy
rm 18.6
- Applied cos-sum to get
\[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
9.2
- Applied associate--l- to get
\[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
9.2
- Using strategy
rm 9.2
- Applied add-cbrt-cube to get
\[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left(\cos \varepsilon\right)}^3}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
9.2
- Applied add-cbrt-cube to get
\[\color{red}{\cos x} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3}} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
9.3
- Applied cbrt-unprod to get
\[\color{red}{\sqrt[3]{{\left(\cos x\right)}^3} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
9.3
- Using strategy
rm 9.3
- Applied add-cube-cbrt to get
\[\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3} - \left(\color{red}{\sin x \cdot \sin \varepsilon} + \cos x\right) \leadsto \sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3} - \left(\color{blue}{{\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3} + \cos x\right)\]
9.4
