Initial program 37.3
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.0
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied add-log-exp22.1
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
- Using strategy
rm Applied tan-quot22.2
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub22.3
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \sin x}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}}\]
Simplified20.8
\[\leadsto \frac{\color{blue}{\left(\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \sin x\right) + \left(\tan \varepsilon \cdot \sin x\right) \cdot \tan x}}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}\]
Simplified20.7
\[\leadsto \frac{\left(\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \sin x\right) + \left(\tan \varepsilon \cdot \sin x\right) \cdot \tan x}{\color{blue}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}}\]
Taylor expanded around inf 0.4
\[\leadsto \frac{\color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}} + \left(\tan \varepsilon \cdot \sin x\right) \cdot \tan x}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}\]
Final simplification0.4
\[\leadsto \frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \left(\tan \varepsilon \cdot \sin x\right) \cdot \tan x}{\cos x - \tan \varepsilon \cdot \left(\tan x \cdot \cos x\right)}\]
