Set up the Lagrangian for a cost minimization problem, then use it to derive the Hicksian demands for goods X and Y when the utility function has the Cobb-Douglas form
U = X^alpha Y^1-alpha
You will do this by finding the levels of X and Y that minimize expenditure PxX+PyY subject to the constraint that utility not fall below
U^0
Substitute those Hicksian demands into the expenditure identity to derive the expenditure function for this Cobb-Douglas individual.
Note that I used the symbol "^" to show exponentiation (but you can use a Word file or whatever works for you).