how do I simplify these trig expressions?

Question

1. cos^-1(cos(47pi/12)
2. arcsin(tan(-pi/6))
3. csc^-1(cos(2017pi/2018))
4. cot(arcsin(1/2 csc(-11pi/4)))

Answer 1

in #1, cosine inverse & cosine function calls, successively, are inverses of one another but the problem is that the innermost argument, pi(47/12), is equivalent to (-pi/12) and that cosine value will be a positive value only slightly less than 1. When the inverse cosine of that positive number is obtained, it will be positive, because the outputs from cosine inverse lie between 0 and pi.  ANSWER: pi/12