#arctan(x)# is the inverse function of #tan(x)#, and it means that, if #y=arctan(x)#, then #y# is a number such that #tan(y)=x#.
In general, #f# is the inverse function of #g# if #f(g(x))=g(f(x))=x#.
On the other hand, #cot(x)# simply is #1/tan(x)#, so it's simply the inverse number of #tan(x)#.
So, you have that, as a function, #arctan(x)# is the inverse of #tan(x)#, which means that composing the two functions results in the identity function. In formulas,
#artan(tan(x))=tan(arctan(x)=x#.
Instead, as a number (i.e. you must fix #x#), #cot(x)# is the inverse of #tan(x)#, which means that multiplying the two numbers gives one as a result:
#tan(x)cot(x)=1# for every #x#.
