How does the lead number of inverse circular function be verified?

    I could put some problem sometimes,today i am learning math,i like math though i can resolve to the simple,around chapter of lead number,I agree that some foundamental equation of the lead number can recite to settle issue,but i reckon the one of inverse circular function is can proved.so i attempt to look for resolution pathway on net.

    like arcsin x:  

derive from y=arcsin x  to get following process;

both use 'sin' in two side of '=' to become siny=x,

computing the lead number:     y' cos y=1

moving position:                          y'=1/cosy,

according cos^2 x+sin^2 x=1:    y'=1/√1-sin^2 y,

    using siny=x to be end:               y'=1/√1-x^2.