Initial program 37.2
\[\tan \left(x + \varepsilon\right) - \tan x\]
Initial simplification37.2
\[\leadsto \tan \left(\varepsilon + x\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.8
\[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
- Using strategy
rm Applied tan-quot21.8
\[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x\]
Applied associate-*r/21.8
\[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \color{blue}{\frac{\tan \varepsilon \cdot \sin x}{\cos x}}} - \tan x\]
Taylor expanded around -inf 21.9
\[\leadsto \color{blue}{\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\right) - \frac{\sin x}{\cos x}}\]
- Using strategy
rm Applied associate--l+12.9
\[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)}\]
Final simplification12.9
\[\leadsto \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\]
