Let the given series be denoted by
E = S +Cnsinnα
Where S=C1sinαcos(n−1)α+c2sin2αcos(n−2)α=..
Writing the above in reverse order , we get
S = Cncosαsin(n−1)α+Cn−1cos2αsin(n−2)α
Adding 2S = (C1+Cn)sin(nα−α+α)+(C2+Cn−1)sin(nα−2α+2α)
or 2S = (C1+C2+C3+...+Cn−1)sinnα
∴sinAcosB+cosAsinB=sin(A+B)+sinnα
Add and subtract (C0÷C1)sinnα
∴2S=(∑ni=0ci)sinnα−(1+1)sinnα
=(2n−2)sinnlα
∴S=(22n−1−1)sinnα
∴E=s+nCnsinnα
=S+sinnα=26n−1sinnα by (2)