How can I show that the amplitude of a reflected wave?

[dam]

In Feynman Lectures on Physics (you can find it online), chapter 33 of volume 1, the author derives Fresnel's formulas for the coefficient of reflection in an unusual way by making considerations about the different possible polarization of light. In this way he derives the squares of the amplitudes of the reflected waves. When it comes to find the amplitude itself he says that it is possible to show by similar arguments that it must be real by considering two light rays coming from both sides of a glass surface simultaneously (he says that it is fun to analyze theoretically, I don't Know what does he mean by fun xD). I tried to use the same arguments to find the equations and then let the amplitude of one of the waves go to 0 to retrieve the original Solution, and I find indeed that the amplitude is the same, although I haven't yet managed to show it is the same derived by Feynman because of the horrible algebra. However I'm far from being sure that I've done everything alright so I wanted to ask if anybody knows How did the author mean to show it. Thank you in advance

 

[Simon Bridge]

Welcome to PF;
"The Author" in question meant for the student to figure it out.
"Horrible algebra" was par for the course in his day - students were expected to be good at algebra, the assignment was supposed to tax their abilities, and he did lecture at a top university.

Instead of trying to figure how Feynman would have done it, why not try seeing how it is usually taught today using a modern textbook?

 

[dam]

Actually I know how it is usually taught (Feynman himself uses the more common approach in the second volume), so it's not a problem of understanding the subject, it's just that I am curious to know how things that everybody think can only be shown by means of Maxwell's equations have actually other explanations.

 

[sophiecentaur]

Actually I know how it is usually taught (Feynman himself uses the more common approach in the second volume), so it's not a problem of understanding the subject, it's just that I am curious to know how things that everybody think can only be shown by means of Maxwell's equations have actually other explanations.

OK but please 'in addition to' ,not 'instead of'. :)