Function y=arccot(x)

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):

1. Domain is all number line.
2. Range is segment (0,pi).
3. Function is neither even, nor odd.
4. Function is decreasing.
5. 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.