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The transfer matrix of the general integrable open XXZ quantum spin chainobeys certain functional relations at roots of unity. By exploiting thesefunctional relations, we determine the Bethe Ansatz solution for the transfermatrix eigenvalues for the special cases that all but one of the boundaryparameters are zero, and the bulk anisotropy parameter is \eta = i\pi/3, i\pi/5,... In an Addendum, these results are extended to the cases that any two ofthe boundary parameters {\alpha_-, \alpha_+,\beta_-, \beta_+} are arbitrary andthe remaining boundary parameters are either \eta or i \pi/2.Comment: 13 pages, LaTeX; amssymb, no figures; v2: published version +  Addendum; v3: correct Eq. (3.40

Bethe ansatz derived from the functional relations of the openXXZchain for new special cases