Initial program 12.0
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Simplified8.4
\[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]
- Using strategy
rm Applied add-sqr-sqrt8.7
\[\leadsto \sin th \cdot \frac{\sin ky}{\color{blue}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}}\]
Applied *-un-lft-identity8.7
\[\leadsto \sin th \cdot \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]
Applied times-frac8.7
\[\leadsto \sin th \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}\]
Applied associate-*r*8.7
\[\leadsto \color{blue}{\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}}\]
- Using strategy
rm Applied div-inv8.8
\[\leadsto \left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}\]
Applied associate-*r*9.8
\[\leadsto \color{blue}{\left(\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \sin ky\right) \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}}\]
- Using strategy
rm Applied pow19.8
\[\leadsto \left(\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \sin ky\right) \cdot \color{blue}{{\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}}\]
Applied pow19.8
\[\leadsto \left(\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \color{blue}{{\left(\sin ky\right)}^{1}}\right) \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\]
Applied pow19.8
\[\leadsto \left(\left(\sin th \cdot \color{blue}{{\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}}\right) \cdot {\left(\sin ky\right)}^{1}\right) \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\]
Applied pow19.8
\[\leadsto \left(\left(\color{blue}{{\left(\sin th\right)}^{1}} \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\right) \cdot {\left(\sin ky\right)}^{1}\right) \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\]
Applied pow-prod-down9.8
\[\leadsto \left(\color{blue}{{\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}} \cdot {\left(\sin ky\right)}^{1}\right) \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\]
Applied pow-prod-down9.8
\[\leadsto \color{blue}{{\left(\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \sin ky\right)}^{1}} \cdot {\left(\frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}\]
Applied pow-prod-down9.8
\[\leadsto \color{blue}{{\left(\left(\left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right) \cdot \sin ky\right) \cdot \frac{1}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\right)}^{1}}\]
Simplified8.4
\[\leadsto {\color{blue}{\left(\frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}\right)}}^{1}\]
Final simplification8.4
\[\leadsto \frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}\]
