Initial program 37.2
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.2
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum22.0
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
Applied associate--l+0.4
\[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}\right)\]
Applied distribute-rgt-out--0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(\cos \varepsilon - 1\right)}\]
- Using strategy
rm Applied flip3--0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\]
Applied associate-*r/0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\]
Simplified0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right)}{\color{blue}{(\left(\cos \varepsilon\right) \cdot \left(1 + \cos \varepsilon\right) + 1)_*}}\]
- Using strategy
rm Applied add-sqr-sqrt0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right)}{\color{blue}{\sqrt{(\left(\cos \varepsilon\right) \cdot \left(1 + \cos \varepsilon\right) + 1)_*} \cdot \sqrt{(\left(\cos \varepsilon\right) \cdot \left(1 + \cos \varepsilon\right) + 1)_*}}}\]
Applied associate-/r*0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right)}{\sqrt{(\left(\cos \varepsilon\right) \cdot \left(1 + \cos \varepsilon\right) + 1)_*}}}{\sqrt{(\left(\cos \varepsilon\right) \cdot \left(1 + \cos \varepsilon\right) + 1)_*}}}\]
Final simplification0.5
\[\leadsto \frac{\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}{\sqrt{(\left(\cos \varepsilon\right) \cdot \left(\cos \varepsilon + 1\right) + 1)_*}}}{\sqrt{(\left(\cos \varepsilon\right) \cdot \left(\cos \varepsilon + 1\right) + 1)_*}} + \cos x \cdot \sin \varepsilon\]
