Initial program 36.1
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied diff-sin36.4
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified14.5
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
Taylor expanded around -inf 14.5
\[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
Simplified14.5
\[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon + x\right)}\]
- Using strategy
rm Applied cos-sum0.3
\[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\]
Taylor expanded around inf 0.3
\[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right)\]
Final simplification0.3
\[\leadsto \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right)\]
