Sin9x+sin10x+sin11x+sin12x=4cosx/2 cosx sin21x/2

(sin9x + sin12x) + (sin10x+sin12x)=4 cos x/2 cosx sin 21/2

2sin(9x+12x)/2 * cos (9x-12x)/2  +  2sin(10x+11x)/2  *cos(10x-11x)/2=4 cos x/2 cosx sin 21/2

2sin2121x/2 * cos3x/2+sin21x/2 * cosx/2=4 cos x/2 cosx sin 21/2

2sin21x/2(cos3x/2 + cosx/2)=4 cos x/2 cosx sin 21/2

2sin21x/2* 2(cos3x/2 +x/2)/2 * cos(3x/2 - x/2)/2= 4 cos x/2 cosx sin 21/2

4sin21x/2 * 2cosx* cos x/2= 4 cos x/2 cosx sin 21/2