Question

Find all solutions in the interval [0, 2π). sec^2x-2= tan^2x

no sol
pi/3
pi/4
pi/6

Answer 1

8 sec^2 x + 4 tan^2 x − 12 = 0 → sec^2x = tan^2x + 1

8 [ tan^2x + 1] + 4tan^2x - 12 = 0

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

12tan^2x - 4 = 0 divide through by 4

3tan^2x - 1 = 0

3tan^2x = 1

tan^2x = 1/3 take the square root of both sides

tanx = ±1 / √3

And on the given interval, this happens at pi/6 , 5pi/6, 7pi/6, 11pi/6

i hope this helps.