Initial program 1.1
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
Initial simplification1.1
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell \cdot 2}{Om} \cdot \frac{\ell \cdot 2}{Om}\right) + 1}} + \frac{1}{2}}\]
- Using strategy
rm Applied associate-*r*0.9
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om}} + 1}} + \frac{1}{2}}\]
- Using strategy
rm Applied *-un-lft-identity0.9
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 \cdot \sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}} + \frac{1}{2}}\]
Applied associate-/r*0.9
\[\leadsto \sqrt{\color{blue}{\frac{\frac{\frac{1}{2}}{1}}{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}} + \frac{1}{2}}\]
Simplified0.9
\[\leadsto \sqrt{\frac{\frac{\frac{1}{2}}{1}}{\color{blue}{\sqrt{\frac{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{Om}{\ell}} \cdot 4}{\frac{Om}{\ell}} + 1}}} + \frac{1}{2}}\]
Final simplification0.9
\[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 + \frac{4 \cdot \frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}}}\]
