Initial program 30.2
\[\frac{1 - \cos x}{x \cdot x}\]
- Using strategy
rm Applied flip--30.3
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]
Applied associate-/l/30.3
\[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}}\]
Simplified15.0
\[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
- Using strategy
rm Applied *-un-lft-identity15.0
\[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot \color{blue}{\left(1 \cdot x\right)}\right) \cdot \left(1 + \cos x\right)}\]
Applied *-un-lft-identity15.0
\[\leadsto \frac{\sin x \cdot \sin x}{\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(1 \cdot x\right)\right) \cdot \left(1 + \cos x\right)}\]
Applied swap-sqr15.0
\[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\left(\left(1 \cdot 1\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + \cos x\right)}\]
Applied associate-*l*15.0
\[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\left(1 \cdot 1\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \cos x\right)\right)}}\]
Applied *-un-lft-identity15.0
\[\leadsto \frac{\color{blue}{1 \cdot \left(\sin x \cdot \sin x\right)}}{\left(1 \cdot 1\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \cos x\right)\right)}\]
Applied times-frac15.0
\[\leadsto \color{blue}{\frac{1}{1 \cdot 1} \cdot \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}}\]
Simplified15.0
\[\leadsto \color{blue}{1} \cdot \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
Simplified0.3
\[\leadsto 1 \cdot \color{blue}{\left(\frac{\frac{\sin x}{x}}{\cos x + 1} \cdot \frac{\sin x}{x}\right)}\]
Final simplification0.3
\[\leadsto \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{x}}{1 + \cos x}\]
