Initial program 31.2
\[\frac{1 - \cos x}{x \cdot x}\]
Initial simplification31.2
\[\leadsto \frac{1 - \cos x}{x \cdot x}\]
- Using strategy
rm Applied flip--31.3
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]
Applied associate-/l/31.3
\[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}}\]
Simplified15.5
\[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
- Using strategy
rm Applied times-frac15.9
\[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}}\]
Simplified15.7
\[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}\]
- Using strategy
rm Applied *-un-lft-identity15.7
\[\leadsto \frac{\color{blue}{1 \cdot \sin x}}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\]
Applied times-frac0.2
\[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \frac{\sin x}{x}\right)} \cdot \tan \left(\frac{x}{2}\right)\]
Applied associate-*l*0.2
\[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)\right)}\]
- Using strategy
rm Applied tan-quot0.2
\[\leadsto \frac{1}{x} \cdot \left(\frac{\sin x}{x} \cdot \color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}\right)\]
Applied associate-*r/0.2
\[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}\]
Applied frac-times0.1
\[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)\right)}{x \cdot \cos \left(\frac{x}{2}\right)}}\]
Simplified0.1
\[\leadsto \frac{\color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{\frac{x}{\sin x}}}}{x \cdot \cos \left(\frac{x}{2}\right)}\]
Final simplification0.1
\[\leadsto \frac{\frac{\sin \left(\frac{x}{2}\right)}{\frac{x}{\sin x}}}{\cos \left(\frac{x}{2}\right) \cdot x}\]
