Uncovered interest rate parity

Hi all

I have to proof that the uncovered interest rate parity is hold if c_2(s)=E_1[c_2].
The two first order conditions are given as:

View attachment 3440

Then I simpliefied this:

View attachment 3441

View attachment 3443

I think, if the Cov-Term equals zero, the UIP is fullfilled. Could it be that the Cov-Term is zero, since our expectations of C_2 are the same as C_2 and thus, the variance of u'(C_2) equals to zero? Or is there another proof that the UIP is fullfilled?

Would be very thankful for your help.
Last edited: