\(\left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right)\)
- Started with
\[\tan \left(x + \varepsilon\right) - \tan x\]
48.9
- Using strategy
rm 48.9
- Applied tan-cotan to get
\[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
49.0
- Using strategy
rm 49.0
- Applied add-cube-cbrt to get
\[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3}\]
49.2
- Applied taylor to get
\[{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
0.3
- Taylor expanded around 0 to get
\[\color{red}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)}\]
0.3
- Applied simplify to get
\[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right)\]
0.3
- Applied final simplification
