> Mike Sebastian maintains a page of “calculator forensics”, recording the result of computing arcsin(arccos(arctan(tan(cos(sin(9)))))) on as many different models of calculator as he or his pals can get their hands on. The value should be exactly 9

That's mathematically not correct. To define the function arcsin you have to choose a codomain, which is typically the interval [-pi/2, pi/2]. So for any values x not in that interval you will have arcsin(sin(x)) != x.