sin^-1(x) = (sinx)^-1 - a common misconception... unless?

quite frustratingly, sin^-1(x) is not equal to (sinx)^-1. but i stumbled upon an idea recently - are there any values of x for which they are equal?

so i took the question to desmos, and

there are two solutions, at about ±0.944. but why?

this problem has created an itch in my brain that not even google can scratch, and i think this could make a great bprp video. if not, i would like to see an in-depth solution at the very least, because this will keep me awake at night until i know why!!