Given first fundamental form, find gaussian curvature

Show that if the first fundamental form with respect to the given parametrization x(u,v) of a surface has the form

du^{2}+2cosw(u,v)dudv+dv^{2} for some function w, then the gaussian curvature K=-w_{uv}/sinw.

Attempt:

E=1

F=cosw(u,v)

G=1

K=eg-f^{2}/EG-F^{2 }I don't know how to find the 2nd fundamental form coefficients with the given information. Or is there another way to solve?

Re: Given first fundamental form, find gaussian curvature

Remember that the curvature depends only on the coefficients of the first fundamental form. There exists a very tedious formula for this.