Find all solutions of the equation in the interval[0, 2π) 3sec^2x−4secx−4=0

Question

Need help on this specific question on my trig hw

Answer 1

let y=secx

3y^2 -4y - 4 =0

factor

(3y+2)(y-2) = 0

set each factor =0 and solve for y

3y+2 = 0, 3y=-2, y =-2/3

y-2 = 0, y=2

secx=2, 1/cosx =2, cosx = 1/2, x=pi/3 or 2pi/3

secx =2/3, 1/cox=2/3, cosx = 3/2 which is impossible, as cosines are never greater than 1

only solutions are x=pi/3 and 2pi/3