Initial program 25.7
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt25.7
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity25.7
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac25.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Applied simplify25.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Applied simplify16.7
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied div-sub16.7
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}\]
- Using strategy
rm Applied div-inv16.7
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \color{blue}{\left(a \cdot d\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}}\right)\]
Applied add-sqr-sqrt16.8
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{c \cdot b}{\color{blue}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}} - \left(a \cdot d\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}\right)\]
Applied times-frac9.5
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\color{blue}{\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}} \cdot \frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}} - \left(a \cdot d\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}\right)\]
Applied prod-diff9.6
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left((\left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(-\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(a \cdot d\right)\right))_* + (\left(-\frac{1}{\sqrt{c^2 + d^2}^*}\right) \cdot \left(a \cdot d\right) + \left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(a \cdot d\right)\right))_*\right)}\]
Applied distribute-lft-in9.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot (\left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(-\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(a \cdot d\right)\right))_* + \frac{1}{\sqrt{c^2 + d^2}^*} \cdot (\left(-\frac{1}{\sqrt{c^2 + d^2}^*}\right) \cdot \left(a \cdot d\right) + \left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(a \cdot d\right)\right))_*}\]
Applied simplify9.5
\[\leadsto \color{blue}{\frac{(\left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(\frac{a}{\sqrt{c^2 + d^2}^*} \cdot \left(-d\right)\right))_*}{\sqrt{c^2 + d^2}^*}} + \frac{1}{\sqrt{c^2 + d^2}^*} \cdot (\left(-\frac{1}{\sqrt{c^2 + d^2}^*}\right) \cdot \left(a \cdot d\right) + \left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(a \cdot d\right)\right))_*\]
Applied simplify1.0
\[\leadsto \frac{(\left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(\frac{a}{\sqrt{c^2 + d^2}^*} \cdot \left(-d\right)\right))_*}{\sqrt{c^2 + d^2}^*} + \color{blue}{\frac{0}{\sqrt{c^2 + d^2}^*}}\]
