We know that sin(pi/6) = 1/2 and cos(pi/3) = 1/2. So arcsin and arccos is defined for x=1/2.
Now csc x = 1/sin x and sec x = 1/cos x.
So if csc x = 1/2, then sin x = 2, which is not in its domain. Same argument applies for sec x. You could think of this in terms of the graph too.
For arctan(x) and arccot(x), think of the graph. It is continuous everywhere and defined. Now one strategy you could use if you do not know of its graph is taking the graph of tan(x) and cot(x), restrict its domain and reflect it to obtain arctan(x) and arccot(x).