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Usually the find_root is pretty useful here - see the example below. However, this particular situation has no (real) roots, I think.

sage: f = cos(cos(cos(cos(x)))) == sin(sin(sin(sin(x))))
sage: f.find_root(-100,100)
------------------------------------------
RuntimeError: f appears to have no zero on the interval

Plotting it makes this clear, plus the obvious periodicity:

sage: plot(f,-100,100)

Sorry - I didn't read the "complex" part! Ignore this answer for now, I'll update shortly. +++++ Usually the find_root is pretty useful here - see the example below. However, this particular situation has no (real) roots, I think.

sage: f = cos(cos(cos(cos(x)))) == sin(sin(sin(sin(x))))
sage: f.find_root(-100,100)
------------------------------------------
RuntimeError: f appears to have no zero on the interval

Plotting it makes this clear, plus the obvious periodicity:

sage: plot(f,-100,100)

Sorry - I didn't read the "complex" part! Ignore this answer for now, I'll update shortly. +++++ now.

+++++

Usually the find_root is pretty useful here - see the example below. However, this particular situation has no (real) roots, I think.

sage: f = cos(cos(cos(cos(x)))) == sin(sin(sin(sin(x))))
sage: f.find_root(-100,100)
------------------------------------------
RuntimeError: f appears to have no zero on the interval

Plotting it makes this clear, plus the obvious periodicity:

sage: plot(f,-100,100)

Sorry - I didn't read the "complex" part! Ignore the answer below.

Even in Maxima this seems to mostly be available for polynomials only, which is also true in Sage. I'd be interested in any answer to this. See my other answer for now.a poor, but cool-looking, substitute.

+++++

Usually the find_root is pretty useful here - see the example below. However, this particular situation has no (real) roots, I think.

sage: f = cos(cos(cos(cos(x)))) == sin(sin(sin(sin(x))))
sage: f.find_root(-100,100)
------------------------------------------
RuntimeError: f appears to have no zero on the interval

Plotting it makes this clear, plus the obvious periodicity:

sage: plot(f,-100,100)