\(\frac{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cot x}\)
- Started with
\[\tan \left(x + \varepsilon\right) - \tan x\]
53.5
- Using strategy
rm 53.5
- Applied tan-cotan to get
\[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
53.5
- Applied tan-quot to get
\[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
53.6
- Applied frac-sub to get
\[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
53.6
- Applied simplify to get
\[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
53.6
- Using strategy
rm 53.6
- Applied cos-sum to get
\[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{red}{\cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
53.4
- Applied associate--r- to get
\[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
52.1
- Using strategy
rm 52.1
- Applied add-cube-cbrt to get
\[\frac{\color{red}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right)} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right)}^3} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
52.3
- Applied taylor to get
\[\frac{{\left(\sqrt[3]{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right)}^3 + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
22.6
- Taylor expanded around 0 to get
\[\frac{\color{red}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
22.6
- Applied simplify to get
\[\frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cot x}\]
22.5
- Applied final simplification
