Initial program 36.7
\[\tan \left(x + \varepsilon\right) - \tan x\]
- 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 add-log-exp21.7
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
- Using strategy
rm Applied tan-quot21.8
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub21.9
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \sin x}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}}\]
Simplified20.5
\[\leadsto \frac{\color{blue}{\tan \varepsilon \cdot \left(\sin x \cdot \tan x\right) + \left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right)}}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}\]
Simplified20.4
\[\leadsto \frac{\tan \varepsilon \cdot \left(\sin x \cdot \tan x\right) + \left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right)}{\color{blue}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}}\]
Taylor expanded around -inf 0.4
\[\leadsto \color{blue}{\frac{\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}}\]
Simplified0.4
\[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x + \frac{\sin x}{\frac{\cos x}{\sin x}}}}}\]
Final simplification0.4
\[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\cos x - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\frac{\sin x}{\frac{\cos x}{\sin x}} + \cos x}}\]
