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Here is something that doesn't answer your question, but could at least help you narrow it down some - and perhaps some code in it could help.

sage: f = cos(cos(cos(cos(x)))) - sin(sin(sin(sin(x))))
sage: complex_plot(f,(-pi,pi),(-pi,pi),plot_points=500)

This will show the results of f on this square. The darker the area, the smaller the modulus (color indicates the argument of the image of that point). You can compare with the identity map to see what is what.

sage: complex_plot(x,(-pi,pi),(-pi,pi),plot_points=500)