Calculating the variance of the ratio of random variables

As others have noted, the formula you provide is incorrect. For general distributions, there is no closed formula for the variance of a ratio. However, you can approximate it by using some Taylor series expansions. The results are presented (in strange notation) in this pdf file.

You can work it out exactly yourself by constructing the Taylor series expansion of $f(X,Y)=X/Y$ about the expected $X$ and $Y$ (E$[X]$ and E$[Y]$). Then look at E$[f(X,Y)]$ and E$[\left(f(X,Y)-\right.$E$\left.[f(X,Y)]\right)^2]$ using those approximations.

The ratio of two random variables does not in general have a well-defined variance, even when the numerator and denominator do. A simple example is the Cauchy distribution which is the ratio of two independent normal random variables. As Sivaram has pointed out in the comments, the formula you have given (with the correction noted by Henry) is for the variance of the difference of two random variables.

Henry's wiki link includes a formula for approximating the ratio distribution with a normal distribution. It seems like this can be used to create the sort of formula you are looking for.