In this paper we show, how a straightforward and natural application of a pair of fundamental identities valid for polynomials orthogonal over the unit circle, can be used to calculate the determinant of the finite Toeplitz matrix, △n=det(wj−k)n−1 j,k=0:=det(∫|z|=1 w(z) zj−k+1 dz 2πi n−1 j,k=0, with the Fisher-Hartwig symbol, w(z)=C(1− z) +iβ(1− 1/z)−iβ, |z|=1, α> −1/2, ∈ R. Here C is the normalisation constant chosen so that w0=12π. We use the same approach to compute a difference equation for expressions related to the determinants of the symbol w(z)=et (z+1/z), a symbol important in the study of random permutations. Finally, we study the analogous equations for the symbol w(z)=etz MПα=1 (z-aα/z), gα.