In this paper, a numerical investigation of power-law fluid flow in the trapezoidal cavity has been conducted by incompressible finite-difference lattice Boltzmann method (IFDLBM). By designing the equilibrium distribution function, the Navier-Stokes equations (NSEs) can be recovered exactly. Through the coordinate transformation method, the body-fitted grid in physical region is transformed into a uniform grid in computational region. The effect of Reynolds $(Re)$ number, the power-law index $n$ and the vertical angle $θ$ on the trapezoidal cavity are investigated. According to the numerical results, we come to some conclusions. For low $Re$ number $Re =100,$ it can be found that the behavior of power-law fluid flow becomes more complicated with the increase of $n.$ And as vertical angle $θ$ decreases, the flow becomes smooth and the number of vortices decreases. For high $Re$ numbers, the flow development becomes more complex, the number and strength of vortices increase. If the Reynolds number increases further, the power-law fluid will changes from steady flow to periodic flow and then to turbulent flow. For the steady flow, the lager the $θ,$ the more complicated the vortices. And the critical $Re$ number from steady to periodic state decreases with the decrease of power-law index $n.$