1. The problem involves simplifying and understanding the transformation properties of operators under rotation represented by $D(R)$ and its adjoint $D^{\dagger}(R)$. 2. The given expression is: $$ D^{\dagger}(R) T_q(k) D(R) = \sum_{q_1} \sum_{q_2} \langle k_1 k_2; q_1 q_2 | k_1 k_2; k q \rangle \times D^{\dagger}(R) X_{q_1}(k_1) D(R) D^{\dagger}(R) Z_{q_2}(k_2) D(R) $$ 3. Using the transformation property of operators under rotation: $$ D^{\dagger}(R) X_{q_1}(k_1) D(R) = \sum_{q'_1} X_{q'_1}(k_1) D_{q'_1 q_1}(R^{-1}) $$ $$ D^{\dagger}(R) Z_{q_2}(k_2) D(R) = \sum_{q'_2} Z_{q'_2}(k_2) D_{q'_2 q_2}(R^{-1}) $$ 4. Substitute these into the original expression: $$ D^{\dagger}(R) T_q(k) D(R) = \sum_{q_1} \sum_{q_2} \sum_{q'_1} \sum_{q'_2} \langle k_1 k_2; q_1 q_2 | k_1 k_2; k q \rangle X_{q'_1}(k_1) D_{q'_1 q_1}(R^{-1}) Z_{q'_2}(k_2) D_{q'_2 q_2}(R^{-1}) $$ 5. Recognize that the product of the $D$ matrices and the Clebsch-Gordan coefficients can be expanded further using completeness relations: $$ = \sum_{q_1} \sum_{q_2} \sum_{q'_1} \sum_{q'_2} \sum_{q''} \sum_{q'} \langle k_1 k_2; q_1 q_2 | k_1 k_2; k q \rangle \times \langle k_1 k_2; q'_1 q'_2 | k_1 k_2; k'' q' \rangle \times \langle k_1 k_2; q_1 q_2 | k_1 k_2; k'' q'' \rangle D_{k''}(R^{-1}) X_{q'_1}(k_1) Z_{q'_2}(k_2) $$ 6. This final expression shows how the operator $T_q(k)$ transforms under rotation $R$ in terms of the rotated operators $X$ and $Z$ and the rotation matrices $D$ acting on the combined indices. 7. The key formula used is the transformation of tensor operators under rotation: $$ D^{\dagger}(R) T_q(k) D(R) = \sum_{q'} T_{q'}(k) D_{q' q}(R^{-1}) $$ 8. This derivation uses the properties of Clebsch-Gordan coefficients and rotation matrices to express the rotated operator in the basis of $X$ and $Z$ operators. Final answer: $$ D^{\dagger}(R) T_q(k) D(R) = \sum_{q_1} \sum_{q_2} \sum_{q'_1} \sum_{q'_2} \sum_{q''} \sum_{q'} \langle k_1 k_2; q_1 q_2 | k_1 k_2; k q \rangle \times \langle k_1 k_2; q'_1 q'_2 | k_1 k_2; k'' q' \rangle \times \langle k_1 k_2; q_1 q_2 | k_1 k_2; k'' q'' \rangle D_{k''}(R^{-1}) X_{q'_1}(k_1) Z_{q'_2}(k_2) $$