sin theta + sin 2 theta + sin 3 theta + .......+ sin n theta = [sin ( n + 1)/2 / sin 0/2 ] multiply by sin n theta / sin 0 /2 dont be confused experts this question is related to pmi so clear my doubts by providing the solution of my query of exact question without providing any link. Share with your friends Share 10 Avinash Garg answered this Dear Student Let P(n) be the statement given byP(n) : sin θ+sin 2θ+sin 3θ+.........+sin nθ = sinn+12θ sin nθ2sin θ2Step 1: We have P(1) : sin θ = sin1+12θ sin 1×θ2sin θ2=sin θ∵ P(1) is true.Step 2: Let P(m) be true. then,P(m) : sin θ+sin 2θ+sin 3θ+.........+sin mθ = sinm+12θ sin mθ2sin θ2 -----(1)we shall now shoe that P(m+1) is true.i.e. P(m+1) : sin θ+sin 2θ+sin 3θ+.........+sin mθ+sin m+1θ = sinm+1+12θ sin m+1θ2sin θ2now, sin θ+sin 2θ+sin 3θ+.........+sin mθ+sin m+1θ = sinm+12θ sin mθ2sin θ2+sin m+1θ (from (1))=sinm+12θ sin mθ2sin θ2+2sin m+12θ cosm+12θ=sin m+12θsin mθ2sin θ2+2cosm+12θ=sin m+12θsin mθ2+2cosm+12θ sin θ2sin θ2=sin m+12θsin mθ2+sin m+22θ-sin mθ2sin θ2= sin m+12θ sin m+22θsin θ2=sin m+12θ sin m+1+12θsin θ2P(m+1) is true.hence, by Principal of mathematical induction P(n) is true for all n∈N Regards 23 View Full Answer
