We can solve the expression $5 \cos(22.5^\circ) \sin(22.5^\circ)$ by utilizing a trigonometric identity. Specifically, the product of sine and cosine can be expressed as:

$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$

Given that we have $\cos(22.5^\circ) \sin(22.5^\circ)$, we can rewrite it as:

$\cos(22.5^\circ) \sin(22.5^\circ) = \frac{1}{2} \sin(45^\circ)$

Since $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, we get:

$\cos(22.5^\circ) \sin(22.5^\circ) = \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4}$

Now, multiply this by 5:

$5 \cos(22.5^\circ) \sin(22.5^\circ) = 5 \times \frac{\sqrt{2}}{4} = \frac{5\sqrt{2}}{4}$

Thus, the value of $5 \cos(22.5^\circ) \sin(22.5^\circ)$ is $\frac{5\sqrt{2}}{4}$.

Would you like further details on this?

Here are some related questions:

1. How do you prove the identity $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$?
2. What are the exact trigonometric values for other common angles, like $30^\circ$ and $60^\circ$?
3. Can you find the value of $\cos(45^\circ)$ and $\sin(45^\circ)$ using the unit circle?
4. How do double-angle identities help in solving trigonometric equations?
5. What are half-angle identities in trigonometry, and how can they be used?

Tip: Memorizing key trigonometric values like $\sin(45^\circ)$, $\cos(30^\circ)$, and their identities makes solving these expressions much faster!