Initial program 12.8
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Simplified9.1
\[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]
- Using strategy
rm Applied add-sqr-sqrt9.4
\[\leadsto \sin th \cdot \frac{\sin ky}{\color{blue}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}}\]
Applied *-un-lft-identity9.4
\[\leadsto \sin th \cdot \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]
Applied times-frac9.4
\[\leadsto \sin th \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}\]
- Using strategy
rm Applied pow19.4
\[\leadsto \sin th \cdot \left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \color{blue}{{\left(\frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}}\right)\]
Applied pow19.4
\[\leadsto \sin th \cdot \left(\color{blue}{{\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}} \cdot {\left(\frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\right)\]
Applied pow-prod-down9.4
\[\leadsto \sin th \cdot \color{blue}{{\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}}\]
Applied pow19.4
\[\leadsto \color{blue}{{\left(\sin th\right)}^{1}} \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\]
Applied pow-prod-down9.4
\[\leadsto \color{blue}{{\left(\sin th \cdot \left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)\right)}^{1}}\]
Simplified9.1
\[\leadsto {\color{blue}{\left(\frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sin th\right)}}^{1}\]
Final simplification9.1
\[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
