Sinx*sin3x+sin4x*sin8x=0

Question

Sinx*sin3x+sin4x*sin8x=0

Answer 1

1/2*(cos(x-3x)-cos(x+3x)+cos(4x-8x)-cos(4x+8x))=
=1/2(cos2x-cos4x+cos4x-cos12x)=1/2(cos2x-cos12x)=1/2*(-2)sin7xsin5x=0
sin7x=0⇒7x=πn⇒x=πn/7
sin5x=0⇒5x=πn⇒x=πn/5