Recognition of roots of unity can be done by looking at the order, then recognizing the pattern of non-zero elements.
Recognition of 2 cos(x) = exp(i x) + exp(-i x) can be done (probably) by conjugating all the terms with negative imaginary part in the cyclotomic, then reducing to normal form, then recognizing a root of unity.
Recognition of roots of unity can be done by looking at the order, then recognizing the pattern of non-zero elements.
Recognition of
2 cos(x) = exp(i x) + exp(-i x)can be done (probably) by conjugating all the terms with negative imaginary part in the cyclotomic, then reducing to normal form, then recognizing a root of unity.