  We demonstrate that constraints on cosmological parameters from the
distribution of clusters as a function of redshift (dN/dz) are complementary to
accurate angular diameter distance (D_A) measurements to clusters, and their
combination significantly tightens constraints on the energy density content of
the Universe. The number counts can be obtained from X-ray and/or SZ
(Sunyaev-Zel'dovich effect) surveys, and the angular diameter distances can be
determined from deep observations of the intra-cluster gas using their thermal
bremsstrahlung X-ray emission and the SZ effect. We combine constraints from
simulated cluster number counts expected from a 12 deg^2 SZ cluster survey and
constraints from simulated angular diameter distance measurements based on the
X-ray/SZ method assuming a statistical accuracy of 10% in the angular diameter
distance determination of 100 clusters with redshifts less than 1.5. We find
that Omega_m can be determined within about 25%, Omega_Lambda within 20%, and w
within 16%. We show that combined dN/dz + D_A constraints can be used to
constrain the different energy densities in the Universe even in the presence
of a few percent redshift dependent systematic error in D_A. We also address
the question of how best to select clusters of galaxies for accurate diameter
distance determinations. We show that the joint dN/dz + D_A constraints on
cosmological parameters for a fixed target accuracy in the energy density
parameters are optimized by selecting clusters with redshift upper cut--offs in
the range 0.5 < z < 1.
