Initial program 37.4
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied diff-sin37.7
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified15.3
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
Taylor expanded around 0 15.3
\[\leadsto 2 \cdot \left(\cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Simplified15.3
\[\leadsto 2 \cdot \left(\cos \color{blue}{\left((\frac{1}{2} \cdot \varepsilon + x)_*\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
- Using strategy
rm Applied fma-udef15.3
\[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Applied cos-sum0.3
\[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
- Using strategy
rm Applied add-log-exp0.4
\[\leadsto 2 \cdot \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\log \left(e^{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Final simplification0.4
\[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
