Summary: We consider a coefficient inverse problem for a parabolic equation in a degenerate angular domain when the moving part of the boundary changes linearly. We show that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with an additional condition. The boundedness of this solution and this additional condition is proved. Moreover, the solution of the considered inverse problem is found in an explicit form and it is proved that the required coefficient is determined uniquely. It is shown that the obtained nontrivial solution of the inverse problem has no singularities and the additional condition also has no singularities.