Initial program 39.8
\[\cos \left(x + \varepsilon\right) - \cos x\]
- Using strategy
rm Applied diff-cos34.3
\[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified15.8
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
- Using strategy
rm Applied log1p-expm1-u15.8
\[\leadsto -2 \cdot \left(\color{blue}{\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Taylor expanded around inf 15.8
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
Simplified15.8
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)\right)}\]
- Using strategy
rm Applied fma-udef15.8
\[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\]
Applied sin-sum0.4
\[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\right)\]
Final simplification0.4
\[\leadsto -2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right)\]
