Question

sinx+sin2x+sin3x+sin4x=0

Answer 1

(sinx+sin4x)+(sin2x+sin3x)=0
2sin(5x/2)cos(3x/2)+2sin(5x/2)cos(x/2)=0
2cos(x/2)*(sin(5x/2)+sin(3x/2))=0
4cos(x/2)*sin(2x)cos(x/2)=0
4cos²(x/2)*sin(2x)=0
cos(x/2)=0
x/2=π/2+πk
x=π+2πk,k∈z
sin2x=0⇒2x=πk⇒x=πk/2,k∈z