To solve the inequality $\cos(x) \geq \sin(x)$ over the interval $-\pi \leq x \leq \pi$, we proceed as follows:

Step 1: Rewrite the inequality

Rearrange the inequality: $\cos(x) - \sin(x) \geq 0$

Factor out $\sqrt{2}$: $\frac{\cos(x) - \sin(x)}{\sqrt{2}} \geq 0$

Step 2: Simplify using trigonometric identity

The left-hand side can be rewritten as: $\frac{\cos(x) - \sin(x)}{\sqrt{2}} = \cos\left(x + \frac{\pi}{4}\right)$

Thus, the inequality becomes: $\cos\left(x + \frac{\pi}{4}\right) \geq 0$

Step 3: Solve $\cos\left(x + \frac{\pi}{4}\right) \geq 0$

The cosine function is non-negative when: $-\frac{\pi}{2} \leq x + \frac{\pi}{4} \leq \frac{\pi}{2}$

Rearrange to solve for $x$:

1. Subtract $\frac{\pi}{4}$ from all sides: $-\frac{\pi}{2} - \frac{\pi}{4} \leq x \leq \frac{\pi}{2} - \frac{\pi}{4}$
2. Simplify the bounds: $-\frac{3\pi}{4} \leq x \leq \frac{\pi}{4}$

Step 4: Final solution in interval notation

The solution to the inequality $\cos(x) \geq \sin(x)$ on $-\pi \leq x \leq \pi$ is: $\boxed{[-\frac{3\pi}{4}, \frac{\pi}{4}]}$

Would you like further details or clarification? Here are some follow-up questions to expand your understanding:

1. How does shifting the angle affect the cosine function?
2. Can this method be used for other trigonometric inequalities? If so, how?
3. What is the geometric interpretation of $\cos(x) \geq \sin(x)$?
4. How would this solution change if the domain were unrestricted?
5. What does the solution tell us about the symmetry of the functions?

Tip: When solving trigonometric inequalities, rewriting the expression using identities often simplifies the process significantly!