In this paper we shall consider certain kinds of singularly perturbed problems described by quasilinear differential equation of second order with small parameter multiplying the highest derivative, and the appropriate boundary conditions, so that the solution displayes boundary layers. The character of the layer is determined by the use of asymptotic behavior of the exact solution out of the layer, where the exact solution is approximated by the solution of the reduced problem. The ressemblance function for the given problem is determined and used for the domain decomposition, so that the standard spectral methods can be applied inside the layer. The spectral approximation for the layer solution upon the layer subinterval is constructed using monotone iterations. The layer subinterval is determined through the numerical layer length which depends on the perturbation parameter and the degree of the chosen truncated orthogonal series used for the spectral approximation. The error estimate is provided by the use of asymptotic behavior of the exact solution at the endpoints of the layer subintervals using the principle of inverse monotonicity. The numerical example is included, showing the high accuracy of the presented method even when a small number of terms is used in the truncated orthogonal series, which is the result of the appropriately determined layer subinterval. The results are tested according to the Chebyshev orthogonal basis.