Question

Is `f(x)=cotx*tanx` increasing or decreasing at `x=pi/3`?

Answer 1

Neither. `f(x)` is a constant function on its domain.

Explanation:

`cotx = 1/tanx`, so for every `x` for which both `tanx` and `cotx` are defined, we have `cotx tanx = 1`

So for all `x` except `pi/2k` where `k` is an integer, we get `f(x)=1` so `f` is neither increasing nor decreasing on any interval.

I don't know what definition you've been given for `f` is increasing at a point, but I cannot imagine a definition that would allow a constant function to be increasing at any point.