Maxwell equations' plane wave reduction

[samgrace]

Hello, I have been attempting to reduce Maxwell's equations to the forms shown in the image, by substituting the plane wave solutions for E and B of the wave equations back into Maxwell's equations, but I cannot find a way.

Anybody know how to do this?

 

[Dale]

That is it. There is nothing more to be done.

When you have a solution and want to show that it is a solution you simply plug it back into the original equations and show that it satisfied those equations. This is what you have done.