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Question

sinnx=nr=0arsinrx where n is an odd natural number, then


A

a0 = 1, a1 = 2n

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B

a0 = 1, a1 = n

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C

a0 = 0, a1 = n

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D

a0 = 0, a1 = -n

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Solution

The correct option is C

a0 = 0, a1 = n


sin nx = Im(einx) = Im[(cosx+isinx)n]

= nC1cosn1x.sinxnC3cosn3x.sin3x+nC5cosn5x.sin5x+...............

Since n is odd, let n = 2λ + 1

nC1(cos2x)λsinxnC3(cos2x)λ1sin3x+..........

nC1(1sin2x)λsinxnC3(1sin2x)λ1sin3x+..........

nC1sinx(nC1nC1.nC3)sin3x+.......

a0=0,a1=n


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