Question

化简cosψ+cos2ψ+cos3ψ+·······+cosnψ

Answer 1

要用到积化和差的公式：
式子乘以2sin(Ψ/2)再除以2sin(Ψ/2)，
(cosψ+cos2ψ+cos3ψ+.......+cosnψ) *2sin(Ψ/2) / [2sin(Ψ/2)]
然后将2sin(Ψ/2)移入括号里，积化和差，发现很多项都消去了，得出以下：
=[sin(3Ψ/2)-sin(Ψ/2)+sin(5Ψ/2)-sin(3Ψ/2)+…+sin((2n+1)Ψ/2)-sin((2n-1)Ψ/2)]/[2sin(Ψ/2)]
=[-sin(Ψ/2)+sin((2n+1)Ψ/2)]/[2sin(Ψ/2)]
=（-1/2+sin((2n+1)Ψ/2) / [2sin(Ψ/2)]