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What is the result of the summation ∑ₖ₌₁¹³ sin ( (π)/(4) + ((k-1)π)/(6)) sin ( (π)/(4) + (kπ)/(6)) ?

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

Correct answer: 2( √(3) - 1)

Solution

The summation involves trigonometric identities and periodicity. Simplifying the terms using product-to-sum formulas leads to the result 2(√3 − 1).

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