Initial program 36.4
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.4
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied frac-2neg21.4
\[\leadsto \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x\]
Applied simplify21.4
\[\leadsto \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{(\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + \left(-1\right))_*}} - \tan x\]
Taylor expanded around -inf 12.6
\[\leadsto \color{blue}{-\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)}\]
- Using strategy
rm Applied add-cbrt-cube12.6
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}}} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
Applied add-cbrt-cube12.6
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \color{blue}{\sqrt[3]{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}}}{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
Applied add-cbrt-cube12.6
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}}{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
Applied cbrt-unprod12.6
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right)}}}{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
Applied cbrt-undiv12.6
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\color{blue}{\sqrt[3]{\frac{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right)}{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}}} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
Applied simplify12.6
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\sqrt[3]{\color{blue}{\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
