Covariance in terms of variance

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Is it true that?

$$ Cov(X, Y) = \frac{Var(X) \; Var(Y)}{Var(X) + Var(Y)} $$

And if so, please prove or explain why this is so.

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No, it does not hold. Namely, if $X=Y$

$$LHS =Var(X)\\ RHS = \frac12 Var(X)$$