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If sin⁻¹(x) + cot⁻¹(1/2) = pi/2, find the value of x.

1. 0
2. 1/sqrt(5)
3. 2/sqrt(5)
4. sqrt(3)/2

Correct answer: 1/sqrt(5)

Solution

cot⁻¹(1/2) = tan⁻¹(2). Let theta = tan⁻¹(2), so tan(theta) = 2 and sin(theta) = 2/sqrt(5), cos(theta) = 1/sqrt(5). Then sin⁻¹(x) = pi/2 - theta, so x = sin(pi/2 - theta) = cos(theta) = 1/sqrt(5).

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