Find $\operatorname{Cov}(\frac{1}{C}, D)$ given that $\operatorname{Cov}(\frac{A}{BC}, D) = X $, where $\operatorname{Cov}$ is the covariance and $\operatorname{Cov}(A, B) = \operatorname{Cov}(A, C) = \operatorname{Cov}(A, D) = \operatorname{Cov}(B, C) = \operatorname{Cov}(B, D) = 0$.
I'm an experimental physicist trying to combine some measurements to determine the covariance between $\frac{1}{C}$ and $D$, but I only know the covariance between $\frac{A}{BC}$ and $D$. $A$ and $B$ are fully uncorrelated and separate factors. I'm not really sure how to do this as I'm not that familiar with the algebra of covariances.
Is it the case that $\operatorname{Cov}(\frac{A}{BC}, D) = \operatorname{Cov}(\frac{1}{C}, D)$ in this case?
Thanks!