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Derivation

1. Initial program 12.9

   \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
2. Using strategy rm

3. Applied div-inv12.9

   \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)} \cdot \sin th\]

4. Applied associate-*l*13.0

   \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)}\]
5. Using strategy rm

6. Applied associate-*r*12.9

   \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th}\]
7. Using strategy rm

8. Applied add-cube-cbrt13.2

   \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\]
9. Using strategy rm

10. Applied associate-*r/13.2

    \[\leadsto \left(\left(\sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\color{blue}{\frac{\sin ky \cdot 1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}\right) \cdot \sin th\]

11. Applied cbrt-div13.1

    \[\leadsto \left(\left(\sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\sin ky \cdot 1}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}\right) \cdot \sin th\]

12. Simplified13.1

    \[\leadsto \left(\left(\sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \frac{\color{blue}{\sqrt[3]{\sin ky}}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sin th\]

13. Final simplification13.1

    \[\leadsto \left(\left(\sqrt[3]{\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}} \cdot \sin ky} \cdot \sqrt[3]{\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}} \cdot \sin ky}\right) \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}}}\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2019005 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))

Details

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