Why do we use black cartridge in printers?

simple.

if you have a black cartridge the result is always black.

mixing all primary colors tends to result in a dark brown/blackish color instead of true black.

if you lack one color in your color cartridge then your brown/blackish color is now some funky shade like magenta. individual color cartridges avoid this but unless you print alot you risk ink drying up and sucking up a whole wad of cash when you need to keep replacing them.

replacing a black cartridge is cheaper than a color cartridge and there is no waste.

etcetera...etcetera
 
Color printers use black cartridges to save money.

Color ink is generally a little more expensive and you need all three colors to make black; cyan, yellow and magenta. Some color cartridges contains all three colors while others use separate cartridges for each individual color.

If you are using a single combo color cartridge, then when any of the colors are used up you need to replace the entire cartridge unless you don't mind funk colors. This is wasteful and can end up being more expensive in the long run. Cheap color printers generally use these types of cartridges.

Individual color cartridges are used in more expensive color printers, if one color gets used up, then simply replace that single color. Printing black means you use all three colors at the same time as opposed to just black ink. Therefore, some color printers use black cartridges to save you money from replacing all three cartridges and black ink is cheaper.