求证：cosx+cos2x+…+cosnx=cosn+12x•sinn2xsinx2

证明：∵2sin
x
2cosnx=sin(
x
2+nx)+sin(
x
2−nx)．
∴2sin
x
2（cosx+cos2x+…+cosnx）=（sin
3x
2−sin
x
2）+(sin
5x
2−
3x
2)+…+(sin
1+2n
2x−sin
1−2n
2x)
=sin
1+2n
2x−sin
x
2
=2cos
n+1
2xsin
n
2x．
∴cos+cos2x+…+cosnx=
cos
n+1
2x•sin
n
2x
sin
x
2．