Initial program 36.6
\[\tan \left(x + \varepsilon\right) - \tan x\]
Initial simplification36.6
\[\leadsto \tan \left(\varepsilon + x\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.9
\[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
- Using strategy
rm Applied div-inv21.9
\[\leadsto \color{blue}{\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
- Using strategy
rm Applied tan-quot22.0
\[\leadsto \left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied associate-*r/22.0
\[\leadsto \color{blue}{\frac{\left(\tan \varepsilon + \tan x\right) \cdot 1}{1 - \tan \varepsilon \cdot \tan x}} - \frac{\sin x}{\cos x}\]
Applied frac-sub22.1
\[\leadsto \color{blue}{\frac{\left(\left(\tan \varepsilon + \tan x\right) \cdot 1\right) \cdot \cos x - \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \sin x}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}}\]
Simplified20.7
\[\leadsto \frac{\color{blue}{\sin x \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right)}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
Taylor expanded around inf 0.4
\[\leadsto \frac{\sin x \cdot \left(\tan x \cdot \tan \varepsilon\right) + \color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
Final simplification0.4
\[\leadsto \frac{\left(\tan \varepsilon \cdot \tan x\right) \cdot \sin x + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
