Initial program 36.9
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.6
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied tan-quot21.7
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon} - \tan x\]
Applied associate-*l/21.7
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]
- Using strategy
rm Applied add-sqr-sqrt43.0
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \color{blue}{\sqrt{\tan x} \cdot \sqrt{\tan x}}\]
Applied flip--43.0
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x} \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}{1 + \frac{\sin x \cdot \tan \varepsilon}{\cos x}}}} - \sqrt{\tan x} \cdot \sqrt{\tan x}\]
Applied associate-/r/43.0
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x} \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \left(1 + \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)} - \sqrt{\tan x} \cdot \sqrt{\tan x}\]
Applied prod-diff43.0
\[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x} \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) \cdot \left(1 + \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) + \left(-\sqrt{\tan x} \cdot \sqrt{\tan x}\right))_* + (\left(-\sqrt{\tan x}\right) \cdot \left(\sqrt{\tan x}\right) + \left(\sqrt{\tan x} \cdot \sqrt{\tan x}\right))_*}\]
Simplified43.0
\[\leadsto \color{blue}{\left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{\tan \varepsilon}{\cos x} \cdot \sin x\right) + \left(\tan x + \tan \varepsilon\right))_*}{(\left(\frac{\sin x}{\cos x} \cdot \left(-\tan \varepsilon\right)\right) \cdot \left(\frac{\tan \varepsilon}{\cos x} \cdot \sin x\right) + 1)_*} - \tan x\right)} + (\left(-\sqrt{\tan x}\right) \cdot \left(\sqrt{\tan x}\right) + \left(\sqrt{\tan x} \cdot \sqrt{\tan x}\right))_*\]
Simplified21.7
\[\leadsto \left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{\tan \varepsilon}{\cos x} \cdot \sin x\right) + \left(\tan x + \tan \varepsilon\right))_*}{(\left(\frac{\sin x}{\cos x} \cdot \left(-\tan \varepsilon\right)\right) \cdot \left(\frac{\tan \varepsilon}{\cos x} \cdot \sin x\right) + 1)_*} - \tan x\right) + \color{blue}{0}\]
Taylor expanded around -inf 21.8
\[\leadsto \color{blue}{\left(\left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)}\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + 0\]
Simplified0.6
\[\leadsto \color{blue}{\left((\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right) + \left(\frac{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_* \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right))_* + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\right)} + 0\]
Final simplification0.6
\[\leadsto (\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right) + \left(\frac{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_* \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right))_* + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\]
