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!

Learn how to solve the trigonometric expression 5cos(22.5°)sin(22.5°) using double-angle identities and exact trigonometric values. This problem helps solidify your understanding of key trigonometric formulas, suitable for students in Grades 10-12.

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