A formula vaguely similar to Sherman and Morrison's If $V$ is a $p\times n$ matrix and $u$ is an $n$-vector, how can I prove the following equality?
$$
\frac{u^TV^T(VV^T)^{-1}Vu}{1-u^TV^T(VV^T)^{-1}Vu} = u^TV^T[V(I-uu^T)V^T]^{-1}Vu.
$$
My line of attack is to recognize the bottom denominator is a scalar, and tried to bring it into the inverse on top. However, I cannot get equivalence, does anyone know how?

This is simple consequence of Sherman-Morrison-Woodbury formula.
By a direct application the Sherman-Morrison-Woodbury formula we have $$
\begin{align}
{\begin{pmatrix} V(1 - uu^T) V^T \end{pmatrix}}^{-1} &= (VV^T-(Vu)(Vu)^T)^{-1}\\
&= (VV^T)^{-1} + \dfrac{(VV^T)^{-1}(Vu)(Vu)^T(VV^T)^{-1}}{1 - u^TV^T(VV^T)^{-1}Vu}
\end{align}
$$
Multiplying the LHS and RHS above first on the left by $(Vu)^T$ and then on the right by $Vu$ we get $u^TV^T{\begin{pmatrix} V(1 - uu^T) V^T \end{pmatrix}}^{-1}Vu = c + \dfrac{c^2}{1-c}$ where $c = u^TV^T(VV^T)^{-1}Vu.$ 
This simplifies to $\dfrac{c}{1-c}$ and the answer follows.
