\(\frac{\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right) + \sin \varepsilon \cdot \sin x}{\cos \left(\varepsilon + x\right) \cdot \cot x}\)
- Started with
\[\tan \left(x + \varepsilon\right) - \tan x\]
53.3
- Using strategy
rm 53.3
- Applied tan-cotan to get
\[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
53.3
- Applied tan-quot to get
\[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
53.3
- Applied frac-sub to get
\[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
53.4
- Applied simplify to get
\[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
53.4
- Using strategy
rm 53.4
- Applied cos-sum to get
\[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{red}{\cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
53.2
- Applied associate--r- to get
\[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
51.9
- Using strategy
rm 51.9
- Applied add-log-exp to get
\[\frac{\color{red}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right)} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\log \left(e^{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right)} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
59.2
- Applied taylor to get
\[\frac{\log \left(e^{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right) + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
22.7
- Taylor expanded around 0 to get
\[\frac{\color{red}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
22.7
- Applied simplify to get
\[\frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right) + \sin \varepsilon \cdot \sin x}{\cos \left(\varepsilon + x\right) \cdot \cot x}\]
22.7
- Applied final simplification
