Find all solutions of the equation in the interval 0 2 Enter

Find all solutions of the equation in the interval [0, 2). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 8 sec^2 x + 4 tan^2 x 12 = 0

Solution

8 sec^2 x + 4 tan^2 x 12 = 0

use the identity : 1+tan^2x = sec^2x

So, 8( 1+ tan^2x) +4tan^2x -12 =0

12tan^2x = 4

tan^2x = 1/3

tanx = 1/sqrt3 , -1/sqrt3

tanx = 1/sqrt3 ----> x= pi/6 , pi+pi/6 = pi/6 , 7pi/6

tanx = -1/sqrt3

x= pi-pi/6 , 2pi -pi/6 = 5pi/6 , 11pi/6

Solution x =pi/6 , 7pi/6 , 5pi/6 , 11pi/6