We have to show that for invertible m x m matrices A and n x n matrices C, as well as arbitrary n x m matrix B, the following identity holds:
$(A^{-1}+B^T C^{-1} B)^{-1}B^TC^{-1} = AB^T(BAB^T+C)^{-1}$
I know that in general the equality $A^{-1}+B^{-1} = (A+B)^{-1}$ does not hold, and I see that $B^T C^{-1} B$ is an m x m matrix - but how can I continue from here? Hints welcome, please do not solve it fully:-)
Hint
Notice what you get when you multiply your equality on both side by $A^{-1}+B^TC^{-1}B$ on the left and by $BAB^T+C$ on the right. You should obtain a trivial identity. From this point you can go back and prove your (awful) formula. Of course we have to suppose that $A^{-1}+B^TC^{-1}B$ and $BAB^T+C$ are indeed invertible matrices.