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The problem in the image reads:

**Exercise 4:**
Solve the equation \(\tan^2(x) = 6 - 2\sec(x)\), where \(0 \leq x \leq 2\pi\).

### Solution:

1. **Step 1: Convert the equation in terms of \(\sin(x)\) and \(\cos(x)\):**

We know that:
- \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)
- \(\sec(x) = \frac{1}{\cos(x)}\)

Thus, the equation becomes:
\[
\left(\frac{\sin(x)}{\cos(x)}\right)^2 = 6 - 2\left(\frac{1}{\cos(x)}\right)
\]
Simplifying:
\[
\frac{\sin^2(x)}{\cos^2(x)} = 6 - \frac{2}{\cos(x)}
\]

2. **Step 2: Multiply both sides by \(\cos^2(x)\) to eliminate the denominators:**
\[
\sin^2(x) = (6\cos^2(x) - 2\cos(x))
\]

3. **Step 3: Use the identity \(\sin^2(x) = 1 - \cos^2(x)\):**
Substituting this into the equation gives:
\[
1 - \cos^2(x) = 6\cos^2(x) - 2\cos(x)
\]

4. **Step 4: Rearrange the equation:**
Move all terms to one side:
\[
1 = 7\cos^2(x) - 2\cos(x)
\]
\[
7\cos^2(x) - 2\cos(x) - 1 = 0
\]

5. **Step 5: Solve the quadratic equation:**
Let \(y = \cos(x)\). The equation becomes:
\[
7y^2 - 2y - 1 = 0
\]
Solve this quadratic using the quadratic formula:
\[
y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(7)(-1)}}{2(7)}
\]
\[
y = \frac{2 \pm \sqrt{4 + 28}}{14} = \frac{2 \pm \sqrt{32}}{14} = \frac{2 \pm 4\sqrt{2}}{14}
\]
Simplifying:
\[
y = \frac{1 \pm 2\sqrt{2}}{7}
\]

So, we have two possible values for \(\cos(x)\):
\[
\cos(x) = \frac{1 + 2\sqrt{2}}{7} \quad \text{or} \quad \cos(x) = \frac{1 - 2\sqrt{2}}{7}
\]

6. **Step 6: Analyze the solutions:**

Since the values of \(\cos(x)\) need to be between -1 and 1 (by definition of cosine), we discard any values that fall outside this range.

Finally, solve for \(x\) within the interval \(0 \leq x \leq 2\pi\) using the accepted values for \(\cos(x)\).

---

Would you like more details on the solution steps or the final values of \(x\)?

### Related questions:
1. How can you solve a quadratic equation in general form?
2. What are the important trigonometric identities used in solving such equations?
3. How do you convert between trigonometric functions and algebraic expressions?
4. What are the methods to solve trigonometric equations with restricted domains?
5. Can we use graphical methods to approximate the solutions for \(\cos(x)\)?

### Tip:
Always check the domain restrictions when solving trigonometric equations, especially when dealing with cosine, sine, or tangent values.

Solve the equation tan^2(x) = 6 - 2sec(x), 0 <= x <= 2π

This page provides a step-by-step solution to the trigonometric equation tan^2(x) = 6 - 2sec(x) for 0 ≤ x ≤ 2π. The solution includes important trigonometric identities, quadratic formula application, and detailed steps to solve for x.

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