An inequality regarding pojection

Question

Let $a$ and $b$ be two normalized vectors in $\mathbb{R}^k$ such that $a^T b \ll 1$ ($T$ is transpose). Define matrix $C$ such that $[a, b, C]$ is full rcolumn rank, and consider a positive definite matrix $D$. Define projection matrix $P_A:=A^T (AA^T)^{−1}A$. Can we say that $$\frac{a^T D^{-1/2}\left(I - P_{D^{-1/2}C}\right)D^{-1/2}b}{ \left\|\left(I - P_{D^{-1/2}C}\right)D^{-1/2}b\right\|_2 \left\|\left(I - P_{D^{-1/2}C}\right)D^{-1/2}a\right\|_2 }\ll 1$$