Initial program 37.2
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.1
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied flip3--22.1
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]
Applied associate-/r/22.1
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]
Simplified22.1
\[\leadsto \frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \color{blue}{\left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\]
Taylor expanded around -inf 22.2
\[\leadsto \color{blue}{\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{3} \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]
Simplified19.9
\[\leadsto \color{blue}{\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \left(\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \frac{\frac{\sin x}{\cos x}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right)\right) - \frac{\sin x}{\cos x}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}\]
Taylor expanded around inf 19.9
\[\leadsto \left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \color{blue}{\left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)}\right)\right) - \frac{\sin x}{\cos x}\right)}\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\]
Simplified0.4
\[\leadsto \left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \color{blue}{\left(\left(\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}{1 - \frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right)}\right)}\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\]
Final simplification0.4
\[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) + \left(\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) + \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\left(\left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)}} - \frac{\sin x}{\cos x}\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)}}\right)\right)\]
