A matrix inverse identity

Question

For matrices $A, B$, I would like to show and understand the intuition behind the following identity $$ (A+B)^{-1} = A^{-1} - (A+B)^{-1} B A^{-1} $$ assuming the inverses exist.

Answer 1

$$ A^{-1}-(A+B)^{-1}BA^{-1} = (I-(A+B)^{-1}B)A^{-1} = (A+B)^{-1}(A+B-B)A^{-1} = (A+B)^{-1}AA^{-1} = (A+B)^{-1} $$

Answer 2

$$(A+B)\big [ A^{-1} - (A+B)^{-1} B A^{-1}\big ] =I + BA^{-1}-BA^{-1}=I$$

Thus $$(A+B)^{-1} = A^{-1} - (A+B)^{-1} B A^{-1}$$

The intuition behind it is probably $$ \frac {1}{a+b} = \frac {1}{a} -\frac {1}{a+b}\frac {b}{a}$$

where a and b are real numbers.