Initial program 36.8
\[\tan \left(x + \varepsilon\right) - \tan x\]
- 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-log-exp21.6
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
- Using strategy
rm Applied tan-quot21.7
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub21.7
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \sin x}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}}\]
Applied simplify21.7
\[\leadsto \frac{\color{blue}{\tan x \cdot \left(\tan \varepsilon \cdot \sin x + \cos x\right) - \left(\sin x - \tan \varepsilon \cdot \cos x\right)}}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}\]
Applied simplify21.7
\[\leadsto \frac{\tan x \cdot \left(\tan \varepsilon \cdot \sin x + \cos x\right) - \left(\sin x - \tan \varepsilon \cdot \cos x\right)}{\color{blue}{\cos x - \left(\cos x \cdot \tan x\right) \cdot \tan \varepsilon}}\]
Taylor expanded around inf 0.4
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot {\left(\sin x\right)}^{2}}{\cos \varepsilon \cdot \cos x} + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}}{\cos x - \left(\cos x \cdot \tan x\right) \cdot \tan \varepsilon}\]
Applied simplify0.4
\[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\cos x + \frac{\sin x}{\frac{\cos x}{\sin x}}\right)}\]
