Initial program 36.7
\[\tan \left(x + \varepsilon\right) - \tan x\]
Initial simplification36.7
\[\leadsto \tan \left(\varepsilon + x\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.6
\[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
- Using strategy
rm Applied add-log-exp21.7
\[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \color{blue}{\log \left(e^{\tan \varepsilon \cdot \tan x}\right)}} - \tan x\]
- Using strategy
rm Applied tan-quot21.8
\[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan \varepsilon \cdot \tan x}\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub21.9
\[\leadsto \color{blue}{\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \left(1 - \log \left(e^{\tan \varepsilon \cdot \tan x}\right)\right) \cdot \sin x}{\left(1 - \log \left(e^{\tan \varepsilon \cdot \tan x}\right)\right) \cdot \cos x}}\]
Simplified20.5
\[\leadsto \frac{\color{blue}{\tan x \cdot \left(\sin x \cdot \tan \varepsilon\right) + \left(\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x\right)}}{\left(1 - \log \left(e^{\tan \varepsilon \cdot \tan x}\right)\right) \cdot \cos x}\]
Simplified20.4
\[\leadsto \frac{\tan x \cdot \left(\sin x \cdot \tan \varepsilon\right) + \left(\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x\right)}{\color{blue}{\cos x - \tan x \cdot \left(\cos x \cdot \tan \varepsilon\right)}}\]
Taylor expanded around -inf 0.4
\[\leadsto \frac{\tan x \cdot \left(\sin x \cdot \tan \varepsilon\right) + \color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}}{\cos x - \tan x \cdot \left(\cos x \cdot \tan \varepsilon\right)}\]
Final simplification0.4
\[\leadsto \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \tan x + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x - \tan x \cdot \left(\cos x \cdot \tan \varepsilon\right)}\]
