The value of `(1-tan^2 1 5^(@))/(1+tan^2 1 5^(@))` is

Options

A. 1
B. `sqrt(3)`
C. `(sqrt(3))/(2)`
D. 2

Explanation

To find the value of the expression \(\frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ}\), we can use the double angle identity for cosine.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ \frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ} \]

2. **Use the Cosine Double Angle Identity**:
Recall the identity:
\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \]
Here, we can let \(\theta = 15^\circ\).

3. **Substitute \(\theta\)**:
Substitute \(15^\circ\) into the identity:
\[ \cos 2(15^\circ) = \frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ} \]

4. **Calculate \(2 \times 15^\circ\)**:
This simplifies to:
\[ \cos 30^\circ \]

5. **Find \(\cos 30^\circ\)**:
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

6. **Conclusion**:
Therefore, we have:
\[ \frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ} = \frac{\sqrt{3}}{2} \]

### Final Answer:
The value of \(\frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ}\) is \(\frac{\sqrt{3}}{2}\).