Initial program 37.0
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.1
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied tan-quot22.2
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub22.2
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
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
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \left(\frac{\sin x \cdot \sin x}{\cos x} + \cos x\right)}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto \frac{\frac{\sin \varepsilon \cdot \left(\frac{\sin x \cdot \sin x}{\cos x} + \cos x\right)}{\color{blue}{1 \cdot \cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
Applied times-frac0.5
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{1} \cdot \frac{\frac{\sin x \cdot \sin x}{\cos x} + \cos x}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
Applied associate-/l*0.5
\[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{1}}{\frac{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}{\frac{\frac{\sin x \cdot \sin x}{\cos x} + \cos x}{\cos \varepsilon}}}}\]
Simplified0.5
\[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\frac{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}{\frac{\frac{\sin x \cdot \sin x}{\cos x} + \cos x}{\cos \varepsilon}}}\]
Final simplification0.5
\[\leadsto \frac{\sin \varepsilon}{\frac{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}{\frac{\frac{\sin x \cdot \sin x}{\cos x} + \cos x}{\cos \varepsilon}}}\]
