In　this　paper,　we　study　the　large-time　behavior　of　weak　solutions　to　the　initial-boundary　problem　arising　in　a　simplified　Ericksen-Leslie　system　for　nonhomogeneous　incompressible　flows　of　nematic　liquid　crystals　with　a　transformation　condition　of　trigonometric　functions　(called　by　trigonometric　condition　for　simplicity)　posed　on　the　initial　direction　field　in　a　bounded　domain　Ω　⊂ℝ2.　We　show　that　the　kinetic　energy　and　direction　field　converge　to　zero　and　an　equilibrium　state,　respectively,　as　time　goes　to　infinity.　Further,　if　the　initial　density　is　away　from　vacuum　and　bounded,　then　the　density,　and　velocity　and　direction　fields　exponential　decay　to　an　equilibrium　state.　In　addition,　we　also　show　that　the　weak　solutions　of　the　corresponding　compressible　flows　converge　an　equilibrium　state.