Initial program 37.1
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.9
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied tan-quot21.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon} - \tan x\]
Applied associate-*l/21.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]
Taylor expanded around -inf 22.1
\[\leadsto \color{blue}{\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\right) - \frac{\sin x}{\cos x}}\]
- Using strategy
rm Applied associate--l+13.1
\[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)}\]
Final simplification13.1
\[\leadsto \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\]
