if x < -2.0926544674404998e-09 or 6.840282922734168 < x

1. Started with
   \[\frac{1 - \cos x}{{x}^2}\]
   1.3
2. Using strategy rm
   1.3
3. Applied flip-- to get
   \[\frac{\color{red}{1 - \cos x}}{{x}^2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{{x}^2}\]
   1.5
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{{x}^2} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{{x}^2}\]
   1.0
5. Using strategy rm
   1.0
6. Applied square-mult to get
   \[\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\color{red}{{x}^2}} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\color{blue}{x \cdot x}}\]
   1.0
7. Applied *-un-lft-identity to get
   \[\frac{\frac{{\left(\sin x\right)}^2}{\color{red}{1 + \cos x}}}{x \cdot x} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{\color{blue}{1 \cdot \left(1 + \cos x\right)}}}{x \cdot x}\]
   1.0
8. Applied *-un-lft-identity to get
   \[\frac{\frac{{\color{red}{\left(\sin x\right)}}^2}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x} \leadsto \frac{\frac{{\color{blue}{\left(1 \cdot \sin x\right)}}^2}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x}\]
   1.0
9. Applied square-prod to get
   \[\frac{\frac{\color{red}{{\left(1 \cdot \sin x\right)}^2}}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x} \leadsto \frac{\frac{\color{blue}{{1}^2 \cdot {\left(\sin x\right)}^2}}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x}\]
   1.0
10. Applied times-frac to get
    \[\frac{\color{red}{\frac{{1}^2 \cdot {\left(\sin x\right)}^2}{1 \cdot \left(1 + \cos x\right)}}}{x \cdot x} \leadsto \frac{\color{blue}{\frac{{1}^2}{1} \cdot \frac{{\left(\sin x\right)}^2}{1 + \cos x}}}{x \cdot x}\]
    1.0
11. Applied times-frac to get
    \[\color{red}{\frac{\frac{{1}^2}{1} \cdot \frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x \cdot x}} \leadsto \color{blue}{\frac{\frac{{1}^2}{1}}{x} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}}\]
    0.5
12. Applied simplify to get
    \[\color{red}{\frac{\frac{{1}^2}{1}}{x}} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x} \leadsto \color{blue}{\frac{1}{x}} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}\]
    0.5

if -2.0926544674404998e-09 < x < 6.840282922734168

1. Started with
   \[\frac{1 - \cos x}{{x}^2}\]
   61.5
2. Applied taylor to get
   \[\frac{1 - \cos x}{{x}^2} \leadsto \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2\]
   0.0
3. Taylor expanded around 0 to get
   \[\color{red}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2} \leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2}\]
   0.0