\(\frac{b}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\)
- Started with
\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
12.8
- Using strategy
rm 12.8
- Applied add-sqr-sqrt to get
\[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
12.8
- Applied simplify to get
\[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
8.2
- Using strategy
rm 8.2
- Applied flip3-- to get
\[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{\color{blue}{\frac{{\left(b \cdot c\right)}^{3} - {\left(a \cdot d\right)}^{3}}{{\left(b \cdot c\right)}^2 + \left({\left(a \cdot d\right)}^2 + \left(b \cdot c\right) \cdot \left(a \cdot d\right)\right)}}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
21.3
- Applied associate-/l/ to get
\[\color{red}{\frac{\frac{{\left(b \cdot c\right)}^{3} - {\left(a \cdot d\right)}^{3}}{{\left(b \cdot c\right)}^2 + \left({\left(a \cdot d\right)}^2 + \left(b \cdot c\right) \cdot \left(a \cdot d\right)\right)}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{{\left(b \cdot c\right)}^{3} - {\left(a \cdot d\right)}^{3}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2 \cdot \left({\left(b \cdot c\right)}^2 + \left({\left(a \cdot d\right)}^2 + \left(b \cdot c\right) \cdot \left(a \cdot d\right)\right)\right)}}\]
21.3
- Applied taylor to get
\[\frac{{\left(b \cdot c\right)}^{3} - {\left(a \cdot d\right)}^{3}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2 \cdot \left({\left(b \cdot c\right)}^2 + \left({\left(a \cdot d\right)}^2 + \left(b \cdot c\right) \cdot \left(a \cdot d\right)\right)\right)} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
8.2
- Taylor expanded around 0 to get
\[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
8.2
- Applied simplify to get
\[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\]
0.0
- Applied final simplification
