sinχ+sin2χ+sin3χ+sin4χ
=sin4χ+sinχ+sin3χ+sin2χ
=sin{(5+3)x/2}+sin{(5-3)x/2}+sin{(5+1)x/2}+sin{(5-1)x/2}
=2sin(5x/2)cos(3x/2) + 2sin(5x/2)cos(x/2)
=2sin(5x/2){cos(3x/2)+cos(x/2)}
=2sin(5x/2){cos{(2+1)x/2)}+cos{(2-1)x/2}
=4*sin(5x/2)*cosx*cos(x/2)