How can arccot of tanx be simplified?

There is not much that can be done by way of simplification.

Suppose arccot(y) = tan(x)

then y = cot[tan(x)]

= 1/tan(tan(x))

Now cot is NOT the inverse of tan, but its reciprocal. So the expression in the first of above equation cannot be simplified further. Similarly tan[tan(x)] is NOT tan(x)*tan(x) = tan2(x)