Solve: sin2x-sin4x+sin6x=0

Question

This question is from trigonometry chapter in mathematics of class 11th.

"Cbse board"

Answer 1

sin2x-sin4x+sin6x=0

let y=2x

siny-sin2y+sin3y=0

siny-sin(y+y)+sin(y+2y)=0

use addition identities

siny-(sinycosy+cosysiny)+sinycos2y+cosysin2y=0

siny-sinycosy-sinycosy+sinycos(y+y)+cosy(sin[y+y])=0

siny-2sinycosy+siny(cosycosy-sinysiny)+cosy(sinycosy+cosysiny)=0

siny-2sinycosy+siny(cos²y-sin²y)+cosy(2sinycosy)=0

siny-2sinycosy+sinycos²y-sinysin²y+2sinycos²y=0

rearrange terms

siny-sinysin²y  -2sinycosy+sinycos²y+2sinycos²y=0

factor

siny(1-sin²y)-siny(2cosy-cos²y+2cos²y)=0

sinycos²y-siny(cos²y+2cosy)=0

sinycos²y-sinycos²y-2sinycosy=0

-2sinycosy=0

sinycosy=0

siny=0, therefore y=0º and y=180º

cosy=0, therefore y=90º and y=270º

but y=2x

2x=0º so x=0º

2x=180º so x=90º

2x=90º so x=45º

2x=270º so x=135º

Answer 2

First of all x = any multiple of π will solve the equation and there may be other solutions.  I will think about it and get back to you.