Initial program 37.1
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.4
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied add-log-exp21.4
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
- Using strategy
rm Applied tan-quot21.5
\[\leadsto \frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub21.6
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \sin x}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \cos x}}\]
Applied simplify21.6
\[\leadsto \frac{\color{blue}{\cos x \cdot \left(\tan \varepsilon + \tan x\right) + \left(\left(\sin x \cdot \tan \varepsilon\right) \cdot \tan x - \sin x\right)}}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \cos x}\]
Applied simplify21.5
\[\leadsto \frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) + \left(\left(\sin x \cdot \tan \varepsilon\right) \cdot \tan x - \sin x\right)}{\color{blue}{\cos x - \left(\tan \varepsilon \cdot \tan x\right) \cdot \cos x}}\]
Taylor expanded around -inf 0.4
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}}{\cos x - \left(\tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
Applied simplify0.4
\[\leadsto \color{blue}{\frac{\cos x + \frac{\sin x}{\frac{\cos x}{\sin x}}}{\cos x - \cos x \cdot \left(\tan \varepsilon \cdot \tan x\right)} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\]
