Initial program 36.9
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.8
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied add-cube-cbrt21.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\right)} \cdot \tan \varepsilon} - \tan x\]
Applied associate-*l*21.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right)}} - \tan x\]
- Using strategy
rm Applied tan-quot22.0
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub22.0
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right)\right) \cdot \cos x}}\]
Simplified20.7
\[\leadsto \frac{\color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \sin x \cdot \left(\left(-{\left(\sqrt[3]{\tan x}\right)}^{3}\right) \cdot \tan \varepsilon\right)}}{\left(1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right)\right) \cdot \cos x}\]
Simplified20.7
\[\leadsto \frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \sin x \cdot \left(\left(-{\left(\sqrt[3]{\tan x}\right)}^{3}\right) \cdot \tan \varepsilon\right)}{\color{blue}{\left(\cos x \cdot \left(-1 \cdot \tan x\right)\right) \cdot \tan \varepsilon + \cos x}}\]
Taylor expanded around inf 0.6
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}} - \sin x \cdot \left(\left(-{\left(\sqrt[3]{\tan x}\right)}^{3}\right) \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \left(-1 \cdot \tan x\right)\right) \cdot \tan \varepsilon + \cos x}\]
Final simplification0.6
\[\leadsto \frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} - \sin x \cdot \left(\left(-{\left(\sqrt[3]{\tan x}\right)}^{3}\right) \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \left(-1 \cdot \tan x\right)\right) \cdot \tan \varepsilon + \cos x}\]
