I am working on some optimization problems, and I am aware of the method of proving that the "angle of incidence equals the angle of reflection" using Fermat's principle and calculus.
However, my textbook suggests that there is a simple way to prove this without calculus, but I'm a little unsure about how to do this. I would appreciate any advice on how to proceed.
$A$ and $B$ are two points on the same side of the mirror. A ray of light from $A$ reflects off the mirror and goes to $B$. If I understand correctly, what is to be proved is this: among all paths going from $A$ to the mirror and then to $B$, the shortest one is the one for which the two angles are equal.
A way to prove that is to consider the ray of light going from $A$ straight through the mirror to the location of the reflection of $B$. The shortest path is a straight line. Think about what that implies about the various angles.