  \cos (   \theta     )   \sin (   \theta     )   

The expression \( \cos(\theta) \sin(\theta) \) can be simplified using trigonometric identities.

One common approach is to use the double-angle identity for sine:

\[
\sin(2\theta) = 2 \sin(\theta) \cos(\theta)
\]

Rearranging this identity, we get:

\[
\sin(\theta) \cos(\theta) = \frac{1}{2} \sin(2\theta)
\]

Therefore,

\[
\cos(\theta) \sin(\theta) = \frac{1}{2} \sin(2\theta)
\]

This is the simplified form of the expression.

Would you like more details or have any questions?

Here are five related questions that build on this concept:

1. How does the double-angle identity for sine relate to other trigonometric identities?
2. What is the double-angle formula for cosine?
3. How can we express \(\cos^2(\theta) - \sin^2(\theta)\) in terms of double angles?
4. How would you integrate \(\cos(\theta) \sin(\theta)\) with respect to \(\theta\)?
5. Can this expression be useful in calculating certain areas or physics problems?

**Tip:** Knowing common trigonometric identities like double- and half-angle formulas can simplify complex trigonometric expressions easily.

Learn how to simplify the expression \(\cos(\theta) \sin(\theta)\) using the double-angle identity. The step-by-step process demonstrates how to rewrite the expression as \(\frac{1}{2} \sin(2\theta)\), an important trigonometric identity. This solution is suitable for high school students studying trigonometry.

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