Question

factorise cos x + cos 3x + cos 5x + cos 7x

Answer 1

CosC + CosD = 2Cos[(C+D)/2]Cos[(C-D)/2] use this formula

Answer 2

cos x + cos 3x=2Cos[(x+3x)/2]Cos[(x-3x)/2]=2*cos2x*cosx
cos 5x + cos 7x=2Cos[(5x+7x)/2]Cos[(5x-7x)/2]=2*cos6x*cosx
add these
cos x + cos 3x + cos 5x + cos 7x=2*cos2x*cosx+2*cos6x*cosx
=2*cosx(cos2x+cos6x)
again apply C & D formula
2*cosx(cos2x+cos6x)=2*cosx*(2Cos[(2x+6x)/2]Cos[(2x-6x)/2])
=2*cosx(cos4x*cos2x)
=2*cosx*cos2x*cos4x