Stochastic approximation algorithm
The issue is possibly that the input signal you have chosen is not persistently exciting.  This means that the signal doesn't "excite" enough modes of the filter in order to be able to accurately estimate its parameters.  Another way to think about it is that it doesn't have enough energy in enough places in the spectrum: just at the frequency of the cosine, whereas white noise has energy everywhere in the spectrum.

From the paper:


  Recent work (e.g. [1–5]) has shown the convergence and robustness properties of adaptive estimation
  and control algorithms to be closely related to persistence of excitation of regression vectors (equivalently
  that they have a sufficient number of spectral lines [5])