Question 769293
Given:
(1) csc(y) = csc(arcsin(1/x)), where I use the equivalent notation arcsin(z) = sin^(-1)z. Comparing the left side to the right side we see that
(2) y = arcsin(1/x) or
(3) sin(y) = 1/x
Now use the identity
(4) csc(y) = 1/sin(y) and (3) we get
(5) csc(y) = 1/(1/x) or
(6) csc(y) = x 
Answer: csc(arcsin(1/x) = x