If : `x = tan25^(@), "then": (tan 155^(@) - tan 115^(@))/(1 + tan 155^(@) * tan 115^(@))=` A)`(1-x^(2))/(2x)` B)`(1+x^(2))/(2x)` C)`(1+x^(2))/(1-x^(2))` D)`(1-x^(2))/(1+x^(2))`

Options

A. `(1-x^(2))/(2x)`
B. `(1+x^(2))/(2x)`
C. `(1+x^(2))/(1-x^(2))`
D. `(1-x^(2))/(1+x^(2))`

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{\tan 155^\circ - \tan 115^\circ}{1 + \tan 155^\circ \tan 115^\circ} \]

Given that \( x = \tan 25^\circ \), we can use the tangent subtraction formula.

### Step-by-Step Solution:

1. **Identify the angles**: We have \( x = \tan 25^\circ \), and we need to evaluate \( \tan 155^\circ \) and \( \tan 115^\circ \).

2. **Use the tangent subtraction formula**: The formula for the tangent of the difference of two angles is:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, we can set \( A = 155^\circ \) and \( B = 115^\circ \).

3. **Calculate \( A - B \)**:
\[ 155^\circ - 115^\circ = 40^\circ \]
Thus, we can rewrite the expression as:
\[ \frac{\tan 155^\circ - \tan 115^\circ}{1 + \tan 155^\circ \tan 115^\circ} = \tan(40^\circ) \]

4. **Convert \( \tan(40^\circ) \) to cotangent**:
We know that:
\[ \tan(40^\circ) = \cot(90^\circ - 40^\circ) = \cot(50^\circ) \]

5. **Express \( \cot(50^\circ) \) in terms of \( \tan(25^\circ) \)**:
We can use the identity \( \cot \theta = \frac{1}{\tan \theta} \):
\[ \cot(50^\circ) = \tan(90^\circ - 50^\circ) = \tan(40^\circ) \]

6. **Use the double angle formula for tangent**:
We can express \( \tan(50^\circ) \) using the double angle formula:
\[ \tan(2x) = \frac{2\tan x}{1 - \tan^2 x} \]
where \( x = 25^\circ \). Thus:
\[ \tan(50^\circ) = \tan(2 \times 25^\circ) = \frac{2\tan 25^\circ}{1 - \tan^2 25^\circ} = \frac{2x}{1 - x^2} \]

7. **Substituting back**:
Therefore, we have:
\[ \tan(40^\circ) = \frac{1 - x^2}{2x} \]

8. **Final result**:
Therefore, the value of the given expression is:
\[ \frac{1 - x^2}{2x} \]

### Conclusion:
The correct option corresponding to our derived expression is:
\[ \text{A) } \frac{1 - x^2}{2x} \]