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n \choose 1 sin (a) + n \choose 2 sin (2a) +...+ n \choose n sin (na)

Source: LIMIT 2020 Cat 2 Objective P18

June 9, 2020
trigonometrySumlimitcalculus

Problem Statement

Evaluate the following sum: (n1)n \choose 1 sin⁡(a)+\sin (a) + (n2)n \choose 2 sin⁡(2a)+...+\sin (2a) +...+ (nn)n \choose n sin⁡(na)\sin (na)
(A) 2ncos⁡n(a2)sin⁡(na2)2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)
(B) 2nsin⁡n(a2)cos⁡(na2)2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)
(C) 2nsin⁡n(a2)sin⁡(na2)2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)
(D) 2ncos⁡n(a2)cos⁡(na2)2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)
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