Initial program 36.1
\[\tan \left(x + \varepsilon\right) - \tan x\]
- 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-sqr-sqrt42.3
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\sqrt{\tan x} \cdot \sqrt{\tan x}}\]
Applied flip--42.3
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \sqrt{\tan x} \cdot \sqrt{\tan x}\]
Applied associate-/r/42.3
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \sqrt{\tan x} \cdot \sqrt{\tan x}\]
Applied prod-diff42.4
\[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\sqrt{\tan x} \cdot \sqrt{\tan x}\right))_* + (\left(-\sqrt{\tan x}\right) \cdot \left(\sqrt{\tan x}\right) + \left(\sqrt{\tan x} \cdot \sqrt{\tan x}\right))_*}\]
Simplified42.4
\[\leadsto \color{blue}{\left(\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right)} + (\left(-\sqrt{\tan x}\right) \cdot \left(\sqrt{\tan x}\right) + \left(\sqrt{\tan x} \cdot \sqrt{\tan x}\right))_*\]
Simplified21.5
\[\leadsto \left(\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right) + \color{blue}{0}\]
- Using strategy
rm Applied *-un-lft-identity21.5
\[\leadsto \left(\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{\color{blue}{1 \cdot 1} - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right) + 0\]
Applied difference-of-squares21.6
\[\leadsto \left(\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{\color{blue}{\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}} - \tan x\right) + 0\]
Applied *-un-lft-identity21.6
\[\leadsto \left(\frac{\color{blue}{1 \cdot (\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}}{\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)} - \tan x\right) + 0\]
Applied times-frac21.6
\[\leadsto \left(\color{blue}{\frac{1}{1 + \tan \varepsilon \cdot \tan x} \cdot \frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\right) + 0\]
Applied fma-neg21.6
\[\leadsto \color{blue}{(\left(\frac{1}{1 + \tan \varepsilon \cdot \tan x}\right) \cdot \left(\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*} + 0\]
Taylor expanded around inf 21.7
\[\leadsto \color{blue}{\left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + 1\right) \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)\right)\right)} + \left(\frac{\sin \varepsilon}{\left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + 1\right) \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left(\left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + 1\right) \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)\right)\right)} + \frac{\sin x}{\cos x \cdot \left(\left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + 1\right) \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)\right)}\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + 0\]
Simplified0.7
\[\leadsto \color{blue}{\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right)\right) + 1 \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right)\right)}\right) + \left(\frac{\sin \varepsilon}{\left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right)\right) + 1 \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right)\right)}\right))_* + \left((\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)}}{1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{\left(1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot (\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_*}\right))_* - \frac{\sin x}{\cos x}\right)\right)} + 0\]
Final simplification0.7
\[\leadsto (\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\left(1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \cos \varepsilon\right) + \left(1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin \varepsilon}{\left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\left(1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \cos \varepsilon\right) + \left(1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_* + \left((\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{\left(1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot (\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_*}\right))_* - \frac{\sin x}{\cos x}\right)\]
