Function y=cot(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=cot(x)` more than once.
However, if we consider interval `(0,pi)` then on this interval function passes horizontal line test. On this interval function is decreasing and takes all values. Therefore, for function `y=cot(x),0<x<pi` there exists inverse function. This function is denoted by `y=text(arccot)(x)` (sometimes also denoted by `y=acot(x)` or `y=cot^(-1)(x)`) and is read as arccotangent.
We can obtain graph of the function `y=text(arccot)(x)` from graph of the function `y=cot(x),0<x<pi` by symmetrically transforming it about line y=x (see figure).
Properties of the function `y=text(arccot)(x)`:
- Domain is all number line.
- Range is segment `(0,pi)`.
- Function is neither even, nor odd.
- Function is decreasing.
- Lines `y=0` and `y=pi` are horizontal asymptotes as `x->+oo` and `x->-oo` respectively.
From above it follows that records `y=text(arccot)(x)` and `x=cot(y),0<y<pi` are equivalent. If we substitute expression for y from first equation into second then we will obtain that `x=cos(text(arccot)(x))`.
Therefore, for any x `cos(text(arccot)(x))=x,0<text(arccot)(x)<pi`.