Initial program 12.8
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Simplified12.8
\[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]
- Using strategy
rm Applied add-cube-cbrt12.9
\[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sqrt[3]{\sin kx}\right)} + \sin ky \cdot \sin ky}}\]
Applied add-cube-cbrt13.0
\[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sqrt[3]{\sin kx}\right)} \cdot \left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sqrt[3]{\sin kx}\right) + \sin ky \cdot \sin ky}}\]
Applied swap-sqr13.0
\[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)\right) \cdot \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)} + \sin ky \cdot \sin ky}}\]
Final simplification13.0
\[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)\right)}}\]
