Evaluate :
`cos ""(pi)/(8) sin ""(pi)/(8)`

Answer

To evaluate \( \cos\left(\frac{\pi}{8}\right) \sin\left(\frac{\pi}{8}\right) \), we can use the trigonometric identity that relates the product of sine and cosine to sine of double angles.

### Step-by-Step Solution:

1. **Use the Identity**: We know from trigonometric identities that:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
Therefore, we can express \( \sin(\theta) \cos(\theta) \) as:
\[ \sin(\theta) \cos(\theta) = \frac{1}{2} \sin(2\theta) \]

2. **Substituting the Values**: In our case, let \( \theta = \frac{\pi}{8} \). Then:
\[ \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right) = \frac{1}{2} \sin\left(2 \cdot \frac{\pi}{8}\right) \]
This simplifies to:
\[ \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right) = \frac{1}{2} \sin\left(\frac{\pi}{4}\right) \]

3. **Evaluate \( \sin\left(\frac{\pi}{4}\right) \)**: We know that:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

4. **Final Calculation**: Now substituting back into our equation:
\[ \cos\left(\frac{\pi}{8}\right) \sin\left(\frac{\pi}{8}\right) = \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \]

5. **Rationalizing the Denominator**: To express this in a more standard form, we can rationalize the denominator:
\[ \frac{1}{2\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{4} \]

Thus, the final answer is:
\[ \cos\left(\frac{\pi}{8}\right) \sin\left(\frac{\pi}{8}\right) = \frac{\sqrt{2}}{4} \]