  The cosine transforms of functions on the unit sphere play an important role
in convex geometry, the Banach space theory, stochastic geometry and other
areas. Their higher-rank generalization to Grassmann manifolds represents an
interesting mathematical object useful for applications. We introduce more
general integral transforms that reveal distinctive features of higher-rank
objects in full generality. We call these new transforms the composite cosine
transforms, by taking into account that their kernels agree with the composite
power function of the cone of positive definite symmetric matrices. We show
that injectivity of the composite cosine transforms can be studied using
standard tools of the Fourier analysis on matrix spaces. In the framework of
this approach, we introduce associated generalized zeta integrals and give new
simple proofs to the relevant functional relations. Our technique is based on
application of the higher-rank Radon transform on matrix spaces.
