Initial program 39.2
\[\cos \left(x + \varepsilon\right) - \cos x\]
- Using strategy
rm Applied diff-cos33.6
\[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified15.1
\[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
Taylor expanded around inf 15.1
\[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
Simplified15.0
\[\leadsto \color{blue}{\sin \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
- Using strategy
rm Applied sin-sum0.4
\[\leadsto \color{blue}{\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
- Using strategy
rm Applied add-cube-cbrt0.5
\[\leadsto \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
Applied associate-*l*0.5
\[\leadsto \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
Final simplification0.5
\[\leadsto \left(\left(\sqrt[3]{\cos x} \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
