I was solving this :
A point charge is placed between two semi infinite conducting plates inclined at an angle of 30 degrees. The number of images is?
The correct answer is 11, which you get by using the concept of number of images formed by two mirrors at an angle. I understand how method of images works for a single plate, but here we are adopting the whole optics concept of two mirrors. How is that valid? What's the proof that it works?
The proof is inherently bound to the uniqueness theorem of the Laplace equation: "if a given potential satisfies the Laplace equation in a region as well as all the boundary conditions specified in it, then said solution is unique". Therefore, if the system of charges you made up is completely equivalent to the one your original system would give place to in the region of interest, then the potential distribution will be the same. If your question is more along the lines of why we can actually substitute a conducting object for a specific distirbution of point charges, then I'd say that is a deeper question which is rarely explained in any detail in uni courses (in my EM lectures, they simply showed us some field lines and assumed it was enough of a demonstration).
Info on the uniqueness theorem: https://farside.ph.utexas.edu/teaching/em/lectures/node37.html#s310
Info on the method of images can be found in any EM textbook. I recommend Gryffiths, but I don't recall this text giving any valid or convincing enough reasons as to why we can substitute conductors by point charges
