Initial program 25.5
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt25.5
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity25.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac25.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified25.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified16.6
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*r/16.5
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied sub-neg16.5
\[\leadsto \frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(b \cdot c + \left(-a \cdot d\right)\right)}}{\sqrt{c^2 + d^2}^*}\]
Applied distribute-rgt-in16.5
\[\leadsto \frac{\color{blue}{\left(b \cdot c\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*} + \left(-a \cdot d\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
Simplified9.0
\[\leadsto \frac{\left(b \cdot c\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*} + \color{blue}{\left(-a\right) \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied associate-*l*0.5
\[\leadsto \frac{\color{blue}{b \cdot \left(c \cdot \frac{1}{\sqrt{c^2 + d^2}^*}\right)} + \left(-a\right) \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
Final simplification0.5
\[\leadsto \frac{a \cdot \frac{-d}{\sqrt{c^2 + d^2}^*} + b \cdot \left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot c\right)}{\sqrt{c^2 + d^2}^*}\]
