To solve the equation:

\[
\sin^2(x) - \cos^2(x) = \frac{1}{2}
\]

we can proceed by recognizing a trigonometric identity and manipulating the equation as follows:

### Step-by-Step Solution:

1. **Rewrite the Equation Using Trigonometric Identity:**

   Recall that:
   \[
   \sin^2(x) - \cos^2(x) = -\cos(2x)
   \]

   This identity simplifies the equation:
   \[
   -\cos(2x) = \frac{1}{2}
   \]

2. **Solve for \(\cos(2x)\):**

   Dividing both sides by \(-1\):
   \[
   \cos(2x) = -\frac{1}{2}
   \]

3. **Find the General Solution for \(\cos(2x) = -\frac{1}{2}\):**

   The cosine function is \(-\frac{1}{2}\) at angles where:
   \[
   2x = \frac{2\pi}{3} + 2k\pi \quad \text{or} \quad 2x = \frac{4\pi}{3} + 2k\pi \quad \text{for integer } k
   \]

4. **Solve for \(x\):**

   Now, divide each equation by 2:
   \[
   x = \frac{\pi}{3} + k\pi \quad \text{or} \quad x = \frac{2\pi}{3} + k\pi
   \]

   Thus, the general solution is:
   \[
   x = \frac{\pi}{3} + k\pi \quad \text{or} \quad x = \frac{2\pi}{3} + k\pi \quad \text{for integer } k
   \]

### Final Answer:

The solutions to the equation are:
\[
x = \frac{\pi}{3} + k\pi \quad \text{or} \quad x = \frac{2\pi}{3} + k\pi \quad \text{where } k \in \mathbb{Z}
\]

Would you like more details on any of these steps, or do you have any questions?

### Related Questions:

1. What are the main trigonometric identities that can simplify similar expressions?
2. How can we solve trigonometric equations involving both \(\sin\) and \(\cos\) terms?
3. How can we verify the solutions using unit circle values?
4. What methods are there to convert a trigonometric equation into an algebraic equation?
5. What are the general solutions for \(\sin(x) = a\) and \(\cos(x) = a\)?

#### Tip:
Always check if you can use trigonometric identities to simplify equations, as they can often transform complex expressions into simpler forms.

Explore the step-by-step solution to solving the trigonometric equation sin^2(x) - cos^2(x) = 1/2, using identities to simplify the expression. This guide covers essential trigonometric concepts, including identities and angle values, ideal for Grades 10-12.

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