Evaluate :
`cos ^(2) ""(pi)/(12) - sin ^(2) ""(pi)/(12)`

Answer

To evaluate the expression \( \cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right) \), we can use the double angle identity for cosine.

### Step-by-Step Solution:

1. **Identify the Expression**:
We have the expression \( \cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right) \).

2. **Use the Cosine Double Angle Identity**:
Recall the identity:
\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \]
Here, let \( \theta = \frac{\pi}{12} \). Therefore, we can rewrite the expression as:
\[ \cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right) = \cos\left(2 \cdot \frac{\pi}{12}\right) \]

3. **Simplify the Angle**:
Calculate \( 2 \cdot \frac{\pi}{12} \):
\[ 2 \cdot \frac{\pi}{12} = \frac{\pi}{6} \]

4. **Evaluate \( \cos\left(\frac{\pi}{6}\right) \)**:
We know that:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

5. **Final Result**:
Therefore, we have:
\[ \cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}}{2} \]

### Conclusion:
The value of \( \cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right) \) is \( \frac{\sqrt{3}}{2} \).

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