I have been solving problems in "The mechanics and thermodynamics of continua" by Gurtin M.E., Fried E., and Anand L. The initial chapters of the book discuss basic tensor algebra of second order tensors.
One of the exercise problem for cofactors $()^{c}$ (section 2.11, problem 8) ask us to prove $$(\mathbf{ST})^{c}=\mathbf{S}^{c}\mathbf{T}^{c}$$
using following definition or representation for cofactor $$\mathbf{A}^{c} = \left[\mathbf{A}^{2}-(tr\mathbf{A})\mathbf{A}+\frac{1}{2}[(tr\mathbf{A})^{2}-tr\mathbf{A}^{2}]\mathbf{I}\right]^{T}$$ where $\mathbf{A}$ is a second order tensor; $(.)^{c}$ represents cofactor of (.); $(.)^{T}$ represents transpose of (.); $tr\mathbf{A}$ is trace of $\mathbf{A}$; $\mathbf{I}$ is the second order identity tensor.
P.S. I am aware of other proofs using $$\mathbf{A^{c}}=det(\mathbf{A})\mathbf{A^{-T}}$$ $$\mathbf{A^{c}}(\mathbf{u}\times\mathbf{v})=\mathbf{Au}\times\mathbf{Av}$$
I have been trying out this problem since last few days but no success. Any help/hint would be great.