Show that in the Bethe approximation the entropy of the Ising lattice at \(T=T_{C}\) is given by the expression

\[
\frac{S_{c}}{N k}=\ln 2+\frac{q}{2} \ln \left(1-\frac{1}{q}\right)-\frac{q(q-2)}{4(q-1)} \ln \left(1-\frac{2}{q}\right) .
\]

Compare this result with the one following from the Bragg-Williams approximation, namely (12.5.20).