Dimensional correctness test with unknown constants (with unknown units)

I want to determine if an expression, with arbitrary constants, is dimensionally correct. The constants essentially have "wildcard" units which one would solve for to make an expression consistent. How can I do this with Unitful.jl?

For example,

- $a_1 x - a_2 y$ , is dimensionally correct, for any dimensions in $x$ and $y$, because $a_1$ and $a_2$ are free parameters.
- $\cos(a_1*x)$ is dimensionally correct, as the $a_1$ can cancel out the dimensions of $x$.
- But, $\cos(a_1*x - y)$ is **not** dimensionally correct if $y$ has units which cannot be cancelled out.
- $\cos(a_1*x) * a_2 - y$ is dimensionally correct, as the $a_2$ can give the result of the $\cos$ the same units as $y$ has.

I am wondering what the best way to approach this with Unitful.jl is?