Initial program 31.3
\[\frac{1 - \cos x}{x \cdot x}\]
- Using strategy
rm Applied flip--31.4
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]
Applied associate-/l/31.4
\[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}}\]
Simplified15.7
\[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
- Using strategy
rm Applied times-frac16.0
\[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}}\]
Simplified15.9
\[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}\]
- Using strategy
rm Applied add-cube-cbrt16.4
\[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}}}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\]
Applied times-frac1.1
\[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{x} \cdot \frac{\sqrt[3]{\sin x}}{x}\right)} \cdot \tan \left(\frac{x}{2}\right)\]
Applied associate-*l*1.1
\[\leadsto \color{blue}{\frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{x} \cdot \left(\frac{\sqrt[3]{\sin x}}{x} \cdot \tan \left(\frac{x}{2}\right)\right)}\]
Final simplification1.1
\[\leadsto \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sqrt[3]{\sin x}}{x}\right)\]
