Initial program 37.7
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.9
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied tan-quot23.0
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub23.1
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
Taylor expanded around inf 0.4
\[\leadsto \frac{\color{blue}{\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
Simplified0.4
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\sin x \cdot \frac{\sin x}{\cos x} + \cos x\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
- Using strategy
rm Applied quot-tan0.3
\[\leadsto \frac{\color{blue}{\tan \varepsilon} \cdot \left(\sin x \cdot \frac{\sin x}{\cos x} + \cos x\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
Final simplification0.3
\[\leadsto \frac{\tan \varepsilon \cdot \left(\sin x \cdot \frac{\sin x}{\cos x} + \cos x\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\]
