  The goal of this paper is to describe the $\alpha$-cosine transform on
functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector
space. in analytic terms as explicitly as possible. We show that for all but
finitely many complex $\alpha$ the $\alpha$-cosine transform is a composition
of the $(\alpha+2)$-cosine transform with an explicitly written (though
complicated) O(n)-invariant differential operator. For all exceptional values
of $\alpha$ except one we interpret the $\alpha$-cosine transform explicitly as
either the Radon transform or composition of two Radon transforms. Explicit
interpretation of the transform corresponding to the last remaining value
$\alpha$, which is $-(min\{i,n-i\}+1)$, is still an open problem.
