sin(2x) - sin(x) = 0
sin(2x) = 2sin(x)cox(x)
Substitute it in the equation above
2sin(x)cos(x) - sin(x) = 0
Take sin(x) as a common factor
sin(x)[2cos(x) - 1] = 0
thats mean
sin(x) = 0 and 2cos(x) - 1 = 0
Let us solve each one
Since sin(x) = 0
Then x = 0, pi , 2 pi
Since 2cos(x) - 1 = 0
Add 1 to both sides
2cos(x) - 1 + 1 = 0 + 1
2cos(x) = 1
Divide both sides by 2
cos(x) = 1/2
If cos is positive then angle x is in 1st quadrant or 4th quadrant
The angle which has cos = 1/2 is pi/3 (60 degrees)
Then x = pi/3
x = 2pi - pi/3
x = 5/3 pi
Since the domain is 0 < x < 2pi
Then we will not take angle 0 and angle 2 pi
So the answer is
pi/3 , pi and 5/3 pi
