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Solve for x in each equation: (i) cos(2 arcsin x) = 1/3; (ii) arccot x + arctan 3 = pi/2; (iii) arctan((x-1)/(x-2)) + arctan((x+1)/(x+2)) = pi/4; (iv) arcsin x + arcsin 2x = 2*pi/3.

1. (i) x = +/- 1/sqrt(3); (ii) x = 3; (iii) x = +/- 1/sqrt(2); (iv) x = 1/2
2. (i) x = +/- sqrt(2/3); (ii) x = 1/3; (iii) x = 1/sqrt(2); (iv) no real solution
3. (i) x = 1/3; (ii) x = 3; (iii) x = 0; (iv) x = sqrt(3)/2
4. (i) x = +/- 1/sqrt(3); (ii) x = 1/3; (iii) x = +/- 1/2; (iv) x = 1/sqrt(7)

Correct answer: (i) x = +/- 1/sqrt(3); (ii) x = 3; (iii) x = +/- 1/sqrt(2); (iv) x = 1/2

Solution

Each part reduces to a standard inverse-trig identity or a tangent-addition step, then to an algebraic equation; always check the principal-value range to discard extraneous roots.

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