将f(x)表示成cosx的多项式

将f(x)表示成cosx的多项式
将f(x)表示成cosx的多项式

将f(x)表示成cosx的多项式
设g(x)=sin(5x/2)-sin(x/2)
=sin(x/2+2x)-sin(x/2)
=[sin(x/2)cos2x+sin2xcos(x/2)]-sin(x/2)
=sin(x/2)[cos2x-1]+sin2xcos(x/2)
=2sin(x/2)(sinx)^2+2sinxcosxcos(x/2)
=2sinx[sin(x/2)sinx+cosxcos(x/2)]
因为cosx=cos-x
所以g(x)=2sinxcos(2x-x/2)=2sinxcos(3x/2)
f(x)=g(x)/[2sin(x/2)]
=2sinxcos(3x/2)/[2sin(x/2)]
=sinxcos(3x/2)/[sin(x/2)]
=2cos(x/2)cos(3x/2)
=2cos(x/2)cos(2x-x/2)
=2cos(x/2[cos2xcos(x/2)+sin2xsin(x/2)]
=2cos(x/2)cos2xcos(x/2)+2cos(x/2)sin2xsin(x/2)
=2cos2x(cos(x/2))^2+sin2xsinx
=cos2x(1+cosx)+2(sinx)^2cosx
=cos2x+cosx[cos2x+2(sinx)^2]
=cos2x+cosx[1-(sinx)^2+(sinx)^2)]
=cos2x+cosx
=2（cosx）^2+cosx-1