Initial program 36.1
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-quot36.1
\[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied tan-sum21.6
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]
Applied frac-sub21.7
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
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
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{\sin x}{\frac{\cos x}{\sin x}} + \cos x\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
- Using strategy
rm Applied associate-/l*0.4
\[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}{\frac{\sin x}{\frac{\cos x}{\sin x}} + \cos x}}}\]
Final simplification0.4
\[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}{\cos x + \frac{\sin x}{\frac{\cos x}{\sin x}}}}\]
