Enquiry about the proof of covariance and variance

Question

Suppose that I set X ∈ {0,1},

How can I prove that COV(X,Y)/VAR(X) =E(Y|X=1)−E(Y|X=0)?

I try thinking of COV(X,Y)=COV(X,E(Y|X)) and divide the term with the conditional variance.Thus, the unknown in the equation is Y.

What I have tried is as follows.

COV(X,E(Y|X))=E(X E(Y|X))-E(X)E(E(Y|X))

For the variance,

VAR(X)=E(X^2)-(E(X))^2

However, when I divided COV with VAR. it seems that I cannot arrange the term to the answer.

Please, Any hints and guidances are very much appreciated.