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