n大于等于1,θ不是π的倍数,求证sinθ+sin3θ+...+sin((2n-1)θ)=sin^2(nθ)/sinθ

题目

n大于等于1,θ不是π的倍数,求证sinθ+sin3θ+...+sin((2n-1)θ)=sin^2(nθ)/sinθ

答案

数学归纳法：
（前面略）
假设当N=n时成立,即sinθ+sin(3θ)+...+sin[(2n-1)θ]=sin^2(nθ)/sinθ
则当N=n+1时
sinθ+sin(3θ)+...+sin{[2(n+1)-1]θ}=sin[(2n+1)θ]+sin^2(nθ)/sinθ
={sinθsin[(2n+1)θ]+sin^2(nθ)}/sinθ
={sinθ[sin(2nθ)cosθ+cos(2nθ)sinθ]+[1-cos(2nθ)]/2}/sinθ
={sin(2θ)sin(2nθ)/2+cos(2nθ)[1-cos(2θ)]/2+1/2-cos(2nθ)/2}/sinθ
=[sin(2θ)sin(2nθ)-cos(2θ)cos(2nθ)+1]/(2sinθ)
=[-cos(2nθ+2θ)+1]/(2sinθ)
={-cos[2(n+1)θ]+1}/(2sinθ)
=2sin^2[(n+1)θ]/(2sinθ)
=sin^2[(n+1)θ]/sinθ
所以当N=n时成立,则当N=n+1时也成立,所以原式成立.