I have a bicycle with rims: Front: Bontrager BCX-1, Rear: Bontrager BCX-2, 32-hole (Original tires = 26 x 2.00" Tioga Psycho KS). Can I run a 26 x 1.75" on these rims?

Yep.

Sure. You can go even narrower if you wanted to, probably down to 1", depending on the tire. The main thing is you do not want the inflated tire to be narrower than the rim itself.

Short answer is almost certainly yes.

Long answer says that the internal width of your rim determines what size tires are appropriate. If you want to find out what's supposed to work, take the bare rim (tire and tube removed) and use a caliper on the space between the two rim sidewalls. Or just measuring carefully with a tape measure is more than good enough.

As a general rule of thumb, if your proposed tire width is somewhere around 1.5x the internal width of your rim, you're good.

For the old standby go/no-go chart, Sheldon Brown delivers: https://sheldonbrown.com/tire-sizing.html#width
Like Sheldon said, that chart is pretty conservative, and lots of people go wider than is claimed to be safe on that chart.

There is also a Schwalbe (tire manufacturer) chart floating around on the web which you can find with your favorite search engine.