Initial program 36.8
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied diff-sin37.1
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified15.0
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
- Using strategy
rm Applied add-log-exp15.3
\[\leadsto 2 \cdot \left(\color{blue}{\log \left(e^{\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Taylor expanded around -inf 15.3
\[\leadsto 2 \cdot \left(\log \left(e^{\color{blue}{\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)}}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Simplified15.2
\[\leadsto 2 \cdot \left(\log \left(e^{\color{blue}{\cos \left(\varepsilon \cdot \frac{1}{2} + x\right)}}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
- Using strategy
rm Applied cos-sum0.6
\[\leadsto 2 \cdot \left(\log \left(e^{\color{blue}{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Applied exp-diff0.6
\[\leadsto 2 \cdot \left(\log \color{blue}{\left(\frac{e^{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x}}{e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}}\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Applied log-div0.6
\[\leadsto 2 \cdot \left(\color{blue}{\left(\log \left(e^{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x}\right) - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Simplified0.4
\[\leadsto 2 \cdot \left(\left(\color{blue}{\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)} - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Final simplification0.4
\[\leadsto 2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
