Initial program 36.9
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.5
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied add-cube-cbrt22.0
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}\]
Applied flip--22.0
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]
Applied associate-/r/22.0
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]
Applied prod-diff22.1
\[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_* + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*}\]
Applied simplify21.9
\[\leadsto \color{blue}{\left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right)} + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*\]
Applied simplify21.5
\[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right) + \color{blue}{0}\]
Taylor expanded around inf 21.7
\[\leadsto \color{blue}{\left(\left(\frac{\sin \varepsilon \cdot {\left(\sin x\right)}^{2}}{\cos \varepsilon \cdot \left({\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{2} \cdot {\left(\sin x\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}\right)\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{2} \cdot {\left(\sin x\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{2} \cdot {\left(\sin x\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}\right)} + \frac{{\left(\sin \varepsilon\right)}^{2} \cdot \sin x}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{2} \cdot {\left(\sin x\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}\right)\right)}\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + 0\]
Applied simplify0.6
\[\leadsto \color{blue}{(\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\color{blue}{1 \cdot \cos x}}\right)\]
Applied *-un-lft-identity0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\color{blue}{1 \cdot \sin x}}{1 \cdot \cos x}\right)\]
Applied times-frac0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \color{blue}{\frac{1}{1} \cdot \frac{\sin x}{\cos x}}\right)\]
Applied *-un-lft-identity0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \frac{\sin x}{\cos x}}{\color{blue}{1 \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)}} - \frac{1}{1} \cdot \frac{\sin x}{\cos x}\right)\]
Applied times-frac0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\color{blue}{\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_*}{1} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}} - \frac{1}{1} \cdot \frac{\sin x}{\cos x}\right)\]
Applied prod-diff0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \color{blue}{\left((\left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_*}{1}\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) + \left(-\frac{\sin x}{\cos x} \cdot \frac{1}{1}\right))_* + (\left(-\frac{\sin x}{\cos x}\right) \cdot \left(\frac{1}{1}\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{1}{1}\right))_*\right)}\]
Applied simplify0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left(\color{blue}{(\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\right) \cdot \left((\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_*\right) + \left(-\frac{\sin x}{\cos x}\right))_*} + (\left(-\frac{\sin x}{\cos x}\right) \cdot \left(\frac{1}{1}\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{1}{1}\right))_*\right)\]
Applied simplify0.6
\[\leadsto (\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right))_* + \left((\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\right) \cdot \left((\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_*\right) + \left(-\frac{\sin x}{\cos x}\right))_* + \color{blue}{0}\right)\]
