Solve a trig equation

Question

solve the equation for the solutions over the interval [0 degrees, 360 degrees)

 

6sec²xtanx=12tanx

 

I am unsure of where to begin.

Answer 1

6sec²x(tanx) = 12tanx

 

6sec²x(tanx) - 12tanx = 0

 

tanx(sec²x - 2) = 0

 

tanx = 0 or sec²x = 2

 

tanx = 0 or cos²x = 1/2

 

If tanx = 0, then x = 0°, 180°, 360°

 

If cos²x = 1/2, then cosx = ±1/√2

 

   If cosx = 1/√2, then x = 45° or 315°

   If cosx = -1/√2, then x = 135° or 225°

 

Solutions in the interval [0°, 360°]:

 

    0°, 180°, 360°, 45°, 135°, 225°, 315°

 

Answer 2

With the restriction tan x ≠ 0, which requires sin x ≠ 0

6 sec² x tan x = 12 tan x

sec² x tan x = 2 tan x

sec² x = 2

1 / sin² x = 2

sin² x = 1 / 2

 

Can you determine the value of x? (There are four!)