Essentially, you want to solve the two non-linear equations x= u+ sin v, y= v+ cos u for u and v in terms of x and y. I doubt that there will be any elementary solution for that.
Right, so you don't actually need the explicit formula.
For example, to find and .
Take the partials of the equations directly (independent variables are ):
Note that and .
Solve this system of linear equations at the given point for .
Similar process for other derivatives.
Note you do need to solve for u and v at the point , and you use these values when evaluating things like sin(u).