I am reading the paper multimodal image registration by maximization of the correlation ratio and maybe I am wrong but I am not sure to understand why this formula is correct

$$ \rho(X, Y)=\frac{\operatorname{Cov}(X, Y)^{2}}{\operatorname{Var}(X) \operatorname{Var}(Y)}=\frac{[E(X Y)-E(X) E(Y)]^{2}}{\left[E\left(X^{2}\right)-E(X)^{2}\right]\left[E\left(Y^{2}\right)-E(Y)^{2}\right]} $$

Then the author prove that

$$ \rho(X, Y)=\cos ^{2} \alpha $$

But is it not supposed to be ?

$$ \rho(X, Y)=\frac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X)} \sqrt{\operatorname{Var}(Y)}} $$

which leads to

$$ \rho(X, Y)=\cos \alpha $$