Initial program 37.5
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.3
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied frac-2neg22.3
\[\leadsto \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x\]
Applied simplify22.2
\[\leadsto \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{(\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + \left(-1\right))_*}} - \tan x\]
Taylor expanded around inf 13.3
\[\leadsto \color{blue}{-\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)}\right)\right)}\]
- Using strategy
rm Applied add-log-exp22.0
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)} + \left(\frac{\sin x}{\cos x} + \color{blue}{\log \left(e^{\frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)}}\right)}\right)\right)\]
Applied add-log-exp15.2
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)} + \left(\color{blue}{\log \left(e^{\frac{\sin x}{\cos x}}\right)} + \log \left(e^{\frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)}}\right)\right)\right)\]
Applied sum-log14.3
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)} + \color{blue}{\log \left(e^{\frac{\sin x}{\cos x}} \cdot e^{\frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)}}\right)}\right)\]
Applied simplify13.3
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)} + \log \color{blue}{\left(e^{\frac{\sin x}{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin x}{1}\right) + \left(-\cos x\right))_*} + \frac{\sin x}{\cos x}}\right)}\right)\]
Applied simplify13.3
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} - 1\right)} + \log \left(e^{\color{blue}{\frac{\sin x}{\cos x} + \frac{\sin x}{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\sin x\right) + \left(-\cos x\right))_*}}}\right)\right)\]
