Sin^2х+sin^2 2х+sin^2 3х+sin^4х=2

Question

Sin^2х+sin^2 2х+sin^2 3х+sin^4х=2

Answer 1

(1-cos2x)/2+(1-cos4x)/2+(1-cos6x)/2+(1-cos8x)/2=2
1-cos2x+1-cos4x+1-cos6x+1-cos8x=4
cos2x+cos4x+cos6x+cos8x=0
2cos5xcos3x+2cos5xcosx=0
2cos5x(cos3x+cosx)=0
2cos5x*2cos2xcosx=0
cos5x=0⇒5x=π/2+πn⇒x=π/10+πn/5
cos2x=0⇒2x=π/2+πn⇒x=π/4+πn/2
cosx=0⇒x=π/2+πn