Let’s solve the equation step by step.

The equation is:

$(\cos(x + \pi))(\cos(x)) + \frac{1}{2} = 0$

Step 1: Simplify $\cos(x + \pi)$

Using the trigonometric identity: $\cos(x + \pi) = -\cos(x),$ we substitute this into the equation: $(-\cos(x))(\cos(x)) + \frac{1}{2} = 0.$

This simplifies to: $-\cos^2(x) + \frac{1}{2} = 0.$

Step 2: Solve for $\cos^2(x)$

Rearrange the equation: $\cos^2(x) = \frac{1}{2}.$

Step 3: Solve for $\cos(x)$

Taking the square root of both sides: $\cos(x) = \pm \frac{\sqrt{2}}{2}.$

Step 4: Find $x$

The general solutions for $\cos(x) = \pm \frac{\sqrt{2}}{2}$ are: $x = \pm \frac{\pi}{4} + 2n\pi \quad \text{and} \quad x = \pm \frac{3\pi}{4} + 2n\pi, \quad \text{where } n \in \mathbb{Z}.$

Step 5: Find the smallest positive $x$

We need the smallest positive $x$. From the solutions:

1. For $\frac{\pi}{4}$, $x = \frac{\pi}{4}$ (smallest positive).
2. For $-\frac{\pi}{4}$, the next positive is $\frac{7\pi}{4}$, which is larger.
3. For $\frac{3\pi}{4}$, $x = \frac{3\pi}{4}$ (next smallest).
4. For $-\frac{3\pi}{4}$, the next positive is $\frac{5\pi}{4}$, which is larger.

The smallest positive $x$ is: $x = \frac{\pi}{4}.$

Let me know if you need additional clarification or further assistance!

Questions to Expand Understanding:

1. How can we visualize this equation using a unit circle?
2. What are the implications of cosine being positive or negative in different quadrants?
3. How does the periodicity of cosine affect the general solution?
4. How would this solution change if $\cos(x)$ were replaced by $\sin(x)$?
5. Can you derive the same solution using a graphing method?

Tip:

Always confirm the quadrant of the solution when solving trigonometric equations to ensure correct values.