I was at a conference the other day, and Peter Phillips made the comment that if we take the Sine or Cosine of a non-stationary time-series, then the Dickey-Fuller test will suggest that the transformed series is stationary. More specifically, this happens if the sample size is large enough.
That got me thinking, and searching, and eventually I came across a paper by Chien-Ho Wang and Robert M. de Jong (see the reference below). Indeed, they establish precisely the result that Peter was referring to.