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Evaluate the sum S = sumₖ₌₁¹³ 1 / [sin(pi/4 + (k-1)*pi/6) * sin(pi/4 + k*pi/6)].

1. 3 - sqrt(3)
2. 2(3 - sqrt(3))
3. 2(sqrt(3) - 1)
4. 2(2 + sqrt(3))

Correct answer: 2(sqrt(3) - 1)

Solution

After telescoping, S = 2[cot(pi/4) - cot(5pi/12)] = 2[1 - (2 - sqrt(3))] = 2(sqrt(3) - 1).

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