Trig help, 2(cos^2)1/2theta - 3costheta = 0?

first, recall the power reduction formula which states that

cos^2(x) = (1/2)(1+cos(2x))

now,

2cos^2(theta/2) - 3cos(theta) = 0

now substitute cos^2(theta/2) = (1/2)(1+cos(theta))

2(1/2)(1+cos(theta)) - 3cos(theta) = 0

1 +cos(theta) - 3cos(theta) = 0

combine like terms and move the negate terms to the other side

1 = 2cos(theta)

1/2 = cos(theta)

cos(theta) = 1/2 when theta = pi/3

remember that cosine take the same value whether the angle is negative or positive

cos(pi/3) = cos(-pi/2) however, your interval for theta restricts it to positive values, however 2pi can be added to the angle to achieve the same angle in the appropriate interval, so

cos(theta) = 1/2 when theta = pi/3 and 2pi - pi/3 (otherwise known as 5pi/3)