<p>In this paper, we propose an efficient Legendre spectral approximation method grounded in a mixed scheme to address fourth-order equations within intricate sectorial domains. Firstly, we present an appropriate affine transformation alongside an auxiliary second-order equation, transforming the intricate sector region into a standardized rectangular domain. Subsequently, we derive an equivalent, coupled second-order system that was associated with the original problem in this standardized rectangular domain. Given the introduction of singularities and variable coefficients by coordinate transformations, we derive the pivotal polar conditions at the origin to guarantee the well-posedness of the problem. Building upon this condition, we define a class of weighted Sobolev spaces, along with their respective approximation spaces, and formulate both the variational formulation and its discrete counterpart for the second-order coupled system. Theoretically, we rigorously establish the well-posedness of both weak solutions and their approximations. Additionally, leveraging Céa’s lemma and the approximation properties of the two-dimensional projection operator, we derive the error estimates. Specifically, we implement the algorithm to address the transmission eigenvalue problem in a two-dimensional intricate sectorial domain, and substantiate its efficacy and spectral precision through an extensive array of numerical examples.</p>