Initial program 39.4
\[\cos \left(x + \varepsilon\right) - \cos x\]
- Using strategy
rm Applied diff-cos34.0
\[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified15.2
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
Taylor expanded around inf 15.2
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
Simplified15.2
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)\right)}\]
- Using strategy
rm Applied fma-udef15.2
\[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\]
Applied sin-sum0.4
\[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\right)\]
- Using strategy
rm Applied distribute-rgt-in0.4
\[\leadsto -2 \cdot \color{blue}{\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
Final simplification0.4
\[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) + \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)\right) \cdot -2\]
