We need to solve the trigonometric equation:

\[
3\sec^2(x) - 4\tan(x) = 2
\]

### Step-by-step Solution:

1. **Rewrite the equation**:  
   Move all terms to one side to set the equation to 0:
   \[
   3\sec^2(x) - 4\tan(x) - 2 = 0
   \]

2. **Use the identity**:  
   Recall the trigonometric identity:
   \[
   \sec^2(x) = 1 + \tan^2(x)
   \]
   Substitute \(\sec^2(x)\) in the equation:
   \[
   3(1 + \tan^2(x)) - 4\tan(x) - 2 = 0
   \]
   Simplify:
   \[
   3 + 3\tan^2(x) - 4\tan(x) - 2 = 0
   \]
   \[
   3\tan^2(x) - 4\tan(x) + 1 = 0
   \]

3. **Solve the quadratic equation**:  
   Let \(u = \tan(x)\), so the equation becomes:
   \[
   3u^2 - 4u + 1 = 0
   \]
   This is a quadratic equation. Use the quadratic formula:
   \[
   u = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(1)}}{2(3)}
   \]
   Simplify:
   \[
   u = \frac{4 \pm \sqrt{16 - 12}}{6}
   \]
   \[
   u = \frac{4 \pm \sqrt{4}}{6}
   \]
   \[
   u = \frac{4 \pm 2}{6}
   \]
   So, the two solutions are:
   \[
   u = \frac{4 + 2}{6} = 1 \quad \text{and} \quad u = \frac{4 - 2}{6} = \frac{1}{3}
   \]

4. **Find \(\tan(x)\)**:  
   Since \(u = \tan(x)\), we have:
   \[
   \tan(x) = 1 \quad \text{or} \quad \tan(x) = \frac{1}{3}
   \]

5. **Find \(x\)**:  
   - For \(\tan(x) = 1\), \(x = \frac{\pi}{4} + n\pi\), where \(n\) is an integer.
   - For \(\tan(x) = \frac{1}{3}\), use the inverse tangent:
     \[
     x = \tan^{-1}\left(\frac{1}{3}\right) + n\pi
     \]
     Numerically, \(\tan^{-1}\left(\frac{1}{3}\right) \approx 0.321\), so the solutions are:
     \[
     x \approx 0.321 + n\pi
     \]

### General Solution:
\[
x = \frac{\pi}{4} + n\pi \quad \text{or} \quad x \approx 0.321 + n\pi
\]
where \(n\) is any integer.

Would you like more details, or do you have any questions?

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### Related Questions:
1. How do you solve trigonometric equations involving multiple identities?
2. What is the general solution for equations involving \(\tan(x)\)?
3. How is the quadratic formula applied to trigonometric problems?
4. Can you solve equations involving other trigonometric functions like \(\sin(x)\) or \(\cos(x)\)?
5. What is the difference between principal and general solutions in trigonometry?

### Tip:
Always check the domain of the trigonometric functions when solving for the general solution to avoid extraneous solutions.

Learn how to solve the trigonometric equation 3sec^2(x) - 4tan(x) = 2 using trigonometric identities and the quadratic formula. This problem involves key trigonometric concepts and is suitable for students in Grades 10-12.

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