Reciprical means 1/, so the reciprical of cos is 1/cos, or the reciprical of sec is cos.
The inverse of a function f(x) is defined to be f-1(x) such that f[f-1(x)]=x (and likewise f-1[f(x)] = x for obvious reasons).
The function which is f-1(x) when f(x)=sec(x) is arcsec(x), since sec[arcsec(x)] = x (for certain ranges of x).