For invertible matrices A and B does the identity:
$$ (A^{-1} + B^{-1})^{-1} = A - A(A+B)^{-1}A $$
hold? My supervisor suggested that they are equal but I haven't been able to prove this and in the matrix cookbook (http://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) there are separate identities for both sides of this equation, but they are not given as equal to each other.
\begin{eqnarray*} A-A(A+B)^{-1}A &=&A-(A+B-B)(A+B)^{-1}A \\ &=&B(A+B)^{-1}A=[A^{-1}(A+B)B^{-1}]^{-1} \\ &=&(A^{-1}+B^{-1})^{-1} \end{eqnarray*}