https://r2.erweima.ai/mathbot/image/1725893362788mk91ra0d.jpg

The equation presented in the image is:

\[
2 \sin^2(x) - 1 = 0
\]

### Step-by-step Solution:

1. **Simplify the Equation:**
   \[
   2 \sin^2(x) = 1
   \]
   Divide both sides by 2:
   \[
   \sin^2(x) = \frac{1}{2}
   \]
   
2. **Solve for \(\sin(x)\):**
   Take the square root of both sides:
   \[
   \sin(x) = \pm \frac{\sqrt{2}}{2}
   \]
   This gives two possible solutions for \(\sin(x)\):
   \[
   \sin(x) = \frac{\sqrt{2}}{2} \quad \text{or} \quad \sin(x) = -\frac{\sqrt{2}}{2}
   \]
   
3. **Find the general solutions:**

   - For \(\sin(x) = \frac{\sqrt{2}}{2}\), the solutions are:
     \[
     x = \frac{\pi}{4} + 2k\pi \quad \text{or} \quad x = \frac{3\pi}{4} + 2k\pi, \quad \text{for } k \in \mathbb{Z}
     \]
   
   - For \(\sin(x) = -\frac{\sqrt{2}}{2}\), the solutions are:
     \[
     x = \frac{5\pi}{4} + 2k\pi \quad \text{or} \quad x = \frac{7\pi}{4} + 2k\pi, \quad \text{for } k \in \mathbb{Z}
     \]
   
4. **Combine the solutions:**
   The complete set of solutions for \(x\) are:
   \[
   x = \frac{\pi}{4} + k\pi \quad \text{or} \quad x = -\frac{\pi}{4} + k\pi \quad \text{for } k \in \mathbb{Z}
   \]

### Comparison with the choices:

- **Option A** lists \(\frac{\pi}{4} + 2k\pi\) and \(-\frac{\pi}{4} + 2k\pi\), which does not match the general solution.
- **Option B** lists \(\frac{\pi}{4} + k\pi\) and \(-\frac{\pi}{4} + k\pi\), which matches the general solution we found.
- **Option C** introduces incorrect coefficients, so it is not correct.

### Correct Answer:
The correct solution is **Option B**.

---

Would you like further clarification on any steps? Here are 5 related questions to deepen your understanding:

1. How would you solve similar equations involving \(\cos(x)\) or \(\tan(x)\)?
2. Can you explain why we consider both positive and negative values of \(\sin(x)\)?
3. How do we derive the general solution for trigonometric equations?
4. What are some other methods to solve trigonometric equations apart from factoring?
5. What is the significance of the parameter \(k\) in the solution?

**Tip:** Always ensure your trigonometric functions are evaluated within the correct interval when solving equations.

Quelle est la solution dans ℝ de l'équation suivante : 2sin²(x) - 1 = 0 ?

Learn how to solve the trigonometric equation 2sin²(x) - 1 = 0 with a detailed step-by-step solution. This problem covers key trigonometric concepts, including the sine function and periodic solutions. Suitable for students in Grades 10-12.

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