Solve `sin^2x+(1)/4sin^2 3x=sinxsin^2 3x`

Explanation

`sin^(2) x+1/4 sin^(2) 3x= sin x sin^(2) 3x`
or `sin^(2) x-sin x sin^(2) 3x+1/4 sin^(2) 3x=0`
or `(sin x-1/2 sin^(2) 3x)^(2) +1/4 sin^(2) 3x (1- sin^(2) 3x)=0`
or `(sin x-1/2 sin^(2) 3x)^(2) +1/4 sin^(2) 3x cos^(2) 3x=0`
or `(sin x 1/2 sin^(2) 3x)^(2) +1/16 sin^(2) 6x =0`
or `sin x-1/2 sin^(2) 3x=0` and `sin 6x=0`
or `2 sin x=sin^(2) 3x and sin 6x =0`
From `sin 6x=0, x=kpi//6, k in Z`
From here, we choose those values which satisfy the equation `2 sin x=sin^(2) 3x`. Now,
`sin^(2) 3 ((k pi)/6)=sin^(2) (kpi)/2={("1, if k is odd"),("0, if k is even"):}}`