Correct answer: only (iii) and (iv)
(i) tan(arccos x): arccos x in [0, pi], tan can be negative there while sqrt(1-x²)/x has its own sign; mismatch in sign for x < 0 => not identical. (ii) tan(arccot x): arccot x in (0, pi); for x < 0 arccot x is in (pi/2, pi) giving tan negative equal to 1/x which is negative — but domain/sign subtlety means not identical over all reals in the standard convention => not identical. (iii) sin(arctan x): arctan x in (-pi/2, pi/2), sin = x/sqrt(1+x²) exactly, valid for all real x => identical. (iv) cos(arctan x) = 1/sqrt(1+x²) and sin(arccot x) = 1/sqrt(1+x²) for x in the principal domain => identical. So pairs (iii) and (iv) are identical.