Thanks! now it makes sense. I used to use ##\sin^{-1}\theta## instead of ##\arcsin\theta## . They just aren't the same, I guess. Basically,I think we use ##\arcsin## for the functional inverse of ##\sin## and ##\csc## for its multiplicative inverse, instead of ##\sin^{-1}##, in order to avoid this confusion. As ##\sin^n## is multiplicative for ##n>0##, I would say that ##\sin^{-n}=\csc^n##. If one needed to write many times the functional iterates and inverses of ##\sin##, I would recommend to use a notation like ##\sin^{[n]}## or ##\sin^{\circ n}##, which I've found in papers dealing with iterated functions.