I came accross the following equality $$ (B-zI)^{-1}-(B+\tau qq^*-zI)^{-1}=\tau(B-zI)^{-1}qq^*(B+\tau qq^*-zI)^{-1} $$ Here, $z=u+iv \in \mathbb{C^+}$, $A$ is a $N\times N$ matrices, $B$ is $N\times N$ Hermitian, $\tau\in \mathbb{R}$, $q\in \mathbb{C}^N$.
Why is the above equility hold?
Hint
Let $$A=(B-zI)^{-1}-(B+\tau qq^*-zI)^{-1}$$
Calculate $$(B-zI)A(B+\tau qq^*-zI)$$