Initial program 37.4
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum22.4
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied frac-2neg22.4
\[\leadsto \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x\]
Applied simplify22.4
\[\leadsto \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{(\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + \left(-1\right))_*}} - \tan x\]
Taylor expanded around -inf 13.3
\[\leadsto \color{blue}{-\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)}\]
- Using strategy
rm Applied div-inv14.1
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \left(\frac{\sin x}{\cos x} + \color{blue}{\sin x \cdot \frac{1}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}}\right)\right)\]
Applied div-inv13.3
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \left(\color{blue}{\sin x \cdot \frac{1}{\cos x}} + \sin x \cdot \frac{1}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]
Applied distribute-lft-out13.3
\[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \color{blue}{\sin x \cdot \left(\frac{1}{\cos x} + \frac{1}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)}\right)\]
