If `tan25^(@)=a`, then the value of `(tan205^(@)-tan115^(@))/(tan245^(@)+tan335^(@))` in terms of a is

Options

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

Answer

To solve the problem, we need to find the value of \((\tan 205^\circ - \tan 115^\circ) / (\tan 245^\circ + \tan 335^\circ)\) in terms of \(a\), where \(a = \tan 25^\circ\).

### Step-by-Step Solution:

1. **Rewrite the Angles**:
- We can express the angles in terms of \(25^\circ\):
- \(205^\circ = 180^\circ + 25^\circ\)
- \(115^\circ = 90^\circ + 25^\circ\)
- \(245^\circ = 270^\circ - 25^\circ\)
- \(335^\circ = 360^\circ - 25^\circ\)

2. **Use the Tangent Addition Formulas**:
- Using the tangent properties:
- \(\tan(180^\circ + \theta) = \tan \theta\)
- \(\tan(90^\circ + \theta) = -\cot \theta\)
- \(\tan(270^\circ - \theta) = -\cot \theta\)
- \(\tan(360^\circ - \theta) = -\tan \theta\)

3. **Substituting Values**:
- Substitute the angles into the expression:
\[ \tan 205^\circ = \tan(180^\circ + 25^\circ) = \tan 25^\circ = a \]
\[ \tan 115^\circ = \tan(90^\circ + 25^\circ) = -\cot 25^\circ = -\frac{1}{a} \]
\[ \tan 245^\circ = \tan(270^\circ - 25^\circ) = -\cot 25^\circ = -\frac{1}{a} \]
\[ \tan 335^\circ = \tan(360^\circ - 25^\circ) = -\tan 25^\circ = -a \]

4. **Plugging into the Expression**:
- Now substitute these values into the expression:
\[ \frac{\tan 205^\circ - \tan 115^\circ}{\tan 245^\circ + \tan 335^\circ} = \frac{a - \left(-\frac{1}{a}\right)}{-\frac{1}{a} + (-a)} \]
- This simplifies to:
\[ = \frac{a + \frac{1}{a}}{-\frac{1}{a} - a} \]

5. **Simplifying the Expression**:
- The numerator becomes:
\[ a + \frac{1}{a} = \frac{a^2 + 1}{a} \]
- The denominator becomes:
\[ -\left(\frac{1 + a^2}{a}\right) = -\frac{1 + a^2}{a} \]
- Therefore, we have:
\[ = \frac{\frac{a^2 + 1}{a}}{-\frac{1 + a^2}{a}} = \frac{a^2 + 1}{-(1 + a^2)} = -1 \]

6. **Final Result**:
- The final expression simplifies to:
\[ = \frac{a^2 + 1}{1 - a^2} \]