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Question

If cr stand by nCr prove that
C1sinαcos(n1)α+C2sin2αcos(n2)α +C3sin3αcos(n3)α +..+Cnsinnα
=2n1sinnα

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Solution

Let the given series be denoted by
E = S +Cnsinnα
Where S=C1sinαcos(n1)α+c2sin2αcos(n2)α=..
Writing the above in reverse order , we get
S = Cncosαsin(n1)α+Cn1cos2αsin(n2)α
Adding 2S = (C1+Cn)sin(nαα+α)+(C2+Cn1)sin(nα2α+2α)
or 2S = (C1+C2+C3+...+Cn1)sinnα
sinAcosB+cosAsinB=sin(A+B)+sinnα
Add and subtract (C0÷C1)sinnα
2S=(ni=0ci)sinnα(1+1)sinnα
=(2n2)sinnlα
S=(22n11)sinnα
E=s+nCnsinnα
=S+sinnα=26n1sinnα by (2)

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