Remember that the curvature depends only on the coefficients of the first fundamental form. There exists a very tedious formula for this.
Show that if the first fundamental form with respect to the given parametrization x(u,v) of a surface has the form
du2+2cosw(u,v)dudv+dv2 for some function w, then the gaussian curvature K=-wuv/sinw.
I don't know how to find the 2nd fundamental form coefficients with the given information. Or is there another way to solve?