  We settle the dual addition formula for continuous $q$-ultraspherical
polynomials as an expansion in terms of special $q$-Racah polynomials for which
the constant term is given by the linearization formula for the continuous
$q$-ultraspherical polynomials. In a second proof we derive the dual addition
formula from the Rahman--Verma addition formula for these polynomials by using
the self-duality of the polynomials. We also consider the limit case of
continuous $q$-Hermite polynomials.
