Initial program 36.8
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.8
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
Taylor expanded around inf 21.9
\[\leadsto \color{blue}{\left(\frac{\sin \varepsilon}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos \varepsilon} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\right) - \frac{\sin x}{\cos x}}\]
Simplified13.2
\[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \left(\frac{\sin x}{\cos x + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)}\]
- Using strategy
rm Applied frac-sub13.2
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \color{blue}{\frac{\sin x \cdot \cos x - \left(\cos x + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos \varepsilon}\right)\right) \cdot \sin x}{\left(\cos x + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos \varepsilon}\right)\right) \cdot \cos x}}\]
Simplified13.2
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \frac{\color{blue}{\sin x \cdot \left(\cos x - \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)}}{\left(\cos x + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos \varepsilon}\right)\right) \cdot \cos x}\]
Simplified13.2
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \frac{\sin x \cdot \left(\cos x - \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)}{\color{blue}{\cos x \cdot \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}\]
- Using strategy
rm Applied flip--13.2
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \frac{\sin x \cdot \color{blue}{\frac{\cos x \cdot \cos x - \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}{\cos x + \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}}{\cos x \cdot \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\]
Simplified0.4
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \frac{\sin x \cdot \frac{\color{blue}{\left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(2 \cdot \cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\cos x + \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\cos x \cdot \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\]
Simplified0.4
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + 1 \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\cos x}\right)} + \frac{\sin x \cdot \frac{\left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(2 \cdot \cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}{\color{blue}{2 \cdot \cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}}{\cos x \cdot \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\]
Final simplification0.4
\[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon + \sin x \cdot \frac{-\sin \varepsilon}{\cos x}} + \frac{\sin x \cdot \frac{\left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\cos x \cdot 2 - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}{\cos x \cdot 2 - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}{\cos x \cdot \left(\cos x - \sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\]
