Summary: A unilateral weighted shift \(A\) is said to be simple if its weight sequence \(\{\alpha_n\}\) satisfies \(\nabla^3(\alpha_n^2)\neq 0\) for all \(n\geq 2\). We prove that if \(A\) and \(B\) are two simple unilateral weighted shifts, then \(A\otimes I + I\otimes B\) is reducible if and only if \(A\) and \(B\) are unitarily equivalent. We also study the reducing subspaces of \(A^k\otimes I + I\otimes B^l\) and give some examples. As an application, we study the reducing subspaces of multiplication operators \(M_{z^k + \alpha w^l}\) on function spaces.