Solve finding all solutions in 0 2 cosx sin2x sinx cosx si

Solve, finding all solutions in [0, 2). cosx sin2x + sinx cosx - sinx = 0

given equation is cosx sin2x + sinx cosx - sinx = 0

sin2x =2sinx cosx

cosx 2sinx cosx + sinx cosx -sinx=0

now we have sinx in all the terms so take sinx common

sinx [ 2cos^2 x +cosx -1 ] =0

this will be true when

sinx = 0 or [ 2cos^2 x +cosx -1 ] =0

when x =0, or 2cos^2 x +2cosx -cosx -1 =0

or 2cosx (cosx +1) -1 (cosx+1) =0

or (2cosx - 1) (cosx+1) =0

(2cosx - 1) (cosx+1) =0 is true

when

(2cosx - 1) = 0 or (cosx+1) =0

2cosx =1 or cosx =-1

cosx =1/2 or x=

x= /3, 2 -/3 or x=

x=/3 , 5/3 or x=

so the solution set is { 0, /3 , , 5/3}