<s>[INST] $(A^{-1}+B^{-1})^{-1}=A-A(A+B)^{-1}A$ 
Let $A, B$ and $A+B$ be invertible matrices. Show that
$(A^{-1}+B^{-1})^{-1}=A-A(A+B)^{-1}A$

Here's my assumption:
\begin{align}
A^{-1} + B^{-1} &= B^{-1} + A^{-1} \\
&= B^{-1}(A + B)A^{-1} \\
\end{align}
Then
\begin{align}
(A^{-1} + B^{-1})^{-1} &= [B^{-1}(A + B)A^{-1}]^{-1} \\
&=A(A+B)^{-1}B
\end{align}
\begin{align}
(A^{-1} + B^{-1})^{-1}B^{-1} &= A(A+B)^{-1}
\end{align}
\begin{align}
-(A^{-1} + B^{-1})^{-1}B^{-1}A &=- A(A+B)^{-1}A
\end{align}
\begin{align}
A-(A^{-1} + B^{-1})^{-1}B^{-1} A&=A- A(A+B)^{-1}A
\end{align}
But I'm stuck. I would be very appreciate if someone could help.
 [/INST] Hint:
$$
A = A(A+B)^{-1}(A+B)
$$
</s>