Initial program 37.5
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.0
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied flip--22.1
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x\]
Applied associate-/r/22.1
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]
Taylor expanded around -inf 22.2
\[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)}\right)\right)\right) - \frac{\sin x}{\cos x}}\]
Simplified0.6
\[\leadsto \color{blue}{\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)}\]
- Using strategy
rm Applied add-cbrt-cube0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \color{blue}{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}\right)} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied add-cbrt-cube0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\color{blue}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}}} \cdot \sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied add-cbrt-cube0.7
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}}}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}} \cdot \sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied cbrt-undiv0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\color{blue}{\sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x}}} \cdot \sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied cbrt-unprod0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied add-cbrt-cube0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \color{blue}{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}\right) \cdot \sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied add-cbrt-cube0.7
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\color{blue}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}}} \cdot \sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right) \cdot \sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied add-cbrt-cube0.7
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}}}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}} \cdot \sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right) \cdot \sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied cbrt-undiv0.7
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \left(\color{blue}{\sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x}}} \cdot \sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right) \cdot \sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied cbrt-unprod0.7
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \color{blue}{\sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} \cdot \sqrt[3]{\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Applied cbrt-unprod0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \color{blue}{\sqrt[3]{\left(\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right) \cdot \left(\frac{\left(\sin x \cdot \sin x\right) \cdot \sin x}{\left(\cos x \cdot \cos x\right) \cdot \cos x} \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)}}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Simplified0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \sqrt[3]{\color{blue}{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right)\right)\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right)\right)\right)}}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
Final simplification0.6
\[\leadsto \frac{\frac{\sin x}{\cos x}}{1 - \sqrt[3]{\left(\left(\frac{\sin x}{\cos x} \cdot \left(\left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \left(\left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\sin x}{\cos x}\right)}} \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\right)\]
