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Determine which pair(s) of functions are identical: (i) y = tan(arccos x) and y = sqrt(1 - x²)/x (ii) y = tan(arccot x) and y = 1/x (iii) y = sin(arctan x) and y = x/sqrt(1 + x²) (iv) y = cos(arctan x) and y = sin(arccot x) Which pairs are identical (have the same domain and values)?

1. only (iii) and (iv)
2. only (i) and (ii)
3. all four pairs
4. only (i), (iii) and (iv)

Correct answer: only (iii) and (iv)

Solution

(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.

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