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Prove the identities: (i) arcsin(cos(arcsin x)) + arccos(sin(arccos x)) = pi/2 for |x| <= 1; (ii) tan(arctan x + arctan y + arctan z) = cot(arccot x + arccot y + arccot z).

1. Both identities are true (proved).
2. Only (i) is true.
3. Only (ii) is true.
4. Neither identity is true.

Correct answer: Both identities are true (proved).

Solution

(i) cos(arcsin x) = sqrt(1-x²) and sin(arccos x) = sqrt(1-x²), call it u. Then arcsin u + arccos u = pi/2 by the complementary identity. (ii) Let A = arctan x + arctan y + arctan z. Since arccot t = pi/2 - arctan t, the sum arccot x + arccot y + arccot z = 3*pi/2 - A. Then cot(3*pi/2 - A) = tan A. So both sides equal tan A, proving the identity.

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