How does scotch tape make frosted glass clear?
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Frosted glass contains a bunch of surface irregularities – basically lots of dents and bumps – which scatter light. That is, instead of light rays passing straight through, like they do through a smooth & flat pane of glass, the bumps and dents throw the light rays “off course” in random directions as the surfaces of the bumps and dents are angled in all different directions. So, what you get from the other side of the glass is a somewhat random mix of light rays, instead of light rays that went in a straight line from their point of origin. So the glass is still translucent – it still lets light through. But because the origin of the light has been somewhat randomized, the glass is no longer transparent – you can’t see exactly what’s behind it.
Scotch tape has glue on one side, and this glue fills in the irregularities in the surface of the frosted glass. Meanwhile, the other side of the tape is flat and smooth. So, when you apply the tape to the frosted glass you’re effectively undoing all of the roughing-up that was done to it, and making it into a sheet of normal glass with a flat, smooth surface.
This works only because the material in the scotch tape happens to have a refractive index that is the same as the glass (or close enough). A surface between materials will bend light unless those surfaces have the same index of refraction. So, if the tape (including glue) and the glass had different indices, then the light wouldn’t pass straight through and it wouldn’t work – the tape wouldn’t be able to “undo” the dents and bumps.
This also means there’s nothing special about scotch tape – any material with a refractive index sufficiently similar to glass can work, if you can make one side of it conform to the irregularities of the frosted glass, while the other side forms a smooth, flat surface. For instance, you could cover the frosted glass with mineral oil to achieve the same effect (though that would probably require the glass to be horizontal w.r.t. gravity so that the layer of mineral oil can sit on top of it).
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