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Trig Equation

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2 Answers

 Just need to convert the Trigonometric function to one function like tanX
  
     Sec2X - tan4 X =- 3
 
     Tan2 X + 1 - Tan4 X = -3
 
      - Tan4 X + Tan2 X + 1 + 3 = 0
     
         TanX - Tan2 X  - 4 = 0
 
        (  Tan X  - 4 ) ( Tan2 X +1 ) =0  
 
         Tan 2 X = 4        Tan2 X +1 = 0  ( Has no acceptable answer.)
 
          Tan X = ± 2         X = 63.4 °        X = - 63.4°
                                     X = 243.7°       X =  153.4°
 
               These are answers between range of  0 < X < 360°
 
               For all possible answer add 2n( 360° ) to each 4 X values.
      
          Note that 3 was changed to -3, for plus 3 there is no real solution.
             

Comments

- Tan4 X + Tan2 X + 1 - 3 = 0 copied from your solution becomes
-tan4x+tan2x-2=0 or tan4x-tan2x+2=0 whose discriminant is -7
 Thank  you , so I changed it to -3, where there is  a solution . We keep both of them. It doesn't seem like the intention of the problem was for no solution. But we keep both of them on. It is a good exercise. 
If you regard the equation as a quadratic for tan2x, since sec2x=-1+tan2x, it is clear that there are no real solutions.  the discriminant for tan4x-tan2x+2 is -7. Perhaps you meant something different.

Comments

But:
        Sin2 X + Cos2X = 1
 
        Sin2 X / Cos2 X + 1 = Sec2 X
         1 + Tan2 X = sec2 X
         So , substitution in 2nd line is correct.
 

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