Initial program 37.1
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.6
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied add-cbrt-cube21.6
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}}} - \tan x\]
Applied add-cbrt-cube21.7
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}} \cdot \sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}} - \tan x\]
Applied cbrt-unprod21.6
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon\right)}}} - \tan x\]
Applied simplify21.6
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}}} - \tan x\]
Taylor expanded around -inf 33.0
\[\leadsto \color{blue}{\left(\frac{\sin x}{\left(1 - {\left({\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}\right)}^{\frac{1}{3}}\right) \cdot \cos x} + \frac{\sin \varepsilon}{\left(1 - {\left({\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}\right)}^{\frac{1}{3}}\right) \cdot \cos \varepsilon}\right) - \frac{\sin x}{\cos x}}\]
Applied simplify12.8
\[\leadsto \color{blue}{\frac{\sin \varepsilon}{\left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos \varepsilon} - \left(\frac{\sin x}{\cos x} - \frac{\frac{\sin x}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}{\cos x}\right)}\]
- Using strategy
rm Applied add-log-exp21.3
\[\leadsto \frac{\sin \varepsilon}{\left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos \varepsilon} - \left(\frac{\sin x}{\cos x} - \color{blue}{\log \left(e^{\frac{\frac{\sin x}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}{\cos x}}\right)}\right)\]
Applied add-log-exp12.8
\[\leadsto \frac{\sin \varepsilon}{\left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos \varepsilon} - \left(\color{blue}{\log \left(e^{\frac{\sin x}{\cos x}}\right)} - \log \left(e^{\frac{\frac{\sin x}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}{\cos x}}\right)\right)\]
Applied diff-log12.8
\[\leadsto \frac{\sin \varepsilon}{\left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos \varepsilon} - \color{blue}{\log \left(\frac{e^{\frac{\sin x}{\cos x}}}{e^{\frac{\frac{\sin x}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}{\cos x}}}\right)}\]
Applied simplify12.8
\[\leadsto \frac{\sin \varepsilon}{\left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos \varepsilon} - \log \color{blue}{\left(e^{\frac{\sin x}{\cos x} - \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}}}\right)}\]
