What is the purpose of the tail

I know it helps monkeys and some animals for defense like crocodiles etc... And some like kangaroos to jump
But why do 4 legged animals like dogs cats large cats have tails

For some animals such as monkeys, kangaroos, cheetahs, etc. it is balance. For others it is a communication tool used to signal. Dogs for example use it thus....
wagging = happy
stiff and erect = aggressive warning
tucked down and in = fear and submission

For others it is used as a fly swatter such as in cattle, bison, and horses (actually they use it for communication as well but to a lesser degree than dogs)

Others such as deer and rabbits, it is predator avoidance. Deer show white to display to predators that the pred. does not need to even attempt to chase them, because the deer can outrun them...saying "look at me, I am healthy, so give up now"
Rabbits use a different technique.....a constant flash of the tail while being chased gets the predator watching it instead of the rabbit itself. Then a quick tuck and the white disappears....and to the predator, so does the rabbit. Then the rabbit takes a sharp turn and the predator overshoots before its eyes can readjust.

Balance.

Balance

& they can be used to convey emotions, like when a dog wags its tail, for example.