Initial program 31.1
\[\frac{1 - \cos x}{x \cdot x}\]
- Using strategy
rm Applied flip--31.2
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]
Applied associate-/l/31.2
\[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}}\]
Simplified15.6
\[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
Taylor expanded around -inf 15.6
\[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \left(\cos x + 1\right)}}\]
Simplified0.3
\[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x + 1}}\]
- Using strategy
rm Applied div-inv0.3
\[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{\cos x + 1}}\]
- Using strategy
rm Applied flip3-+0.3
\[\leadsto \left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{\color{blue}{\frac{{\left(\cos x\right)}^{3} + {1}^{3}}{\cos x \cdot \cos x + \left(1 \cdot 1 - \cos x \cdot 1\right)}}}\]
Applied associate-/r/0.3
\[\leadsto \left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right) \cdot \color{blue}{\left(\frac{1}{{\left(\cos x\right)}^{3} + {1}^{3}} \cdot \left(\cos x \cdot \cos x + \left(1 \cdot 1 - \cos x \cdot 1\right)\right)\right)}\]
Applied associate-*r*0.3
\[\leadsto \color{blue}{\left(\left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{{\left(\cos x\right)}^{3} + {1}^{3}}\right) \cdot \left(\cos x \cdot \cos x + \left(1 \cdot 1 - \cos x \cdot 1\right)\right)}\]
Simplified0.3
\[\leadsto \left(\left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{{\left(\cos x\right)}^{3} + {1}^{3}}\right) \cdot \color{blue}{\left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right)}\]
Final simplification0.3
\[\leadsto \left(\frac{1}{{\left(\cos x\right)}^{3} + 1} \cdot \left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right)\right) \cdot \left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right)\]
