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 evaluate the expression:

\[ \frac{\tan 205^\circ - \tan 115^\circ}{\tan 245^\circ + \tan 335^\circ} \]

given that \(\tan 25^\circ = a\).

### Step 1: Rewrite the angles using trigonometric identities

We can rewrite the angles in terms of \( \tan 25^\circ \):

1. **For \(\tan 205^\circ\)**:
\[ \tan 205^\circ = \tan(180^\circ + 25^\circ) = \tan 25^\circ = a \]

2. **For \(\tan 115^\circ\)**:
\[ \tan 115^\circ = \tan(90^\circ + 25^\circ) = -\cot 25^\circ = -\frac{1}{\tan 25^\circ} = -\frac{1}{a} \]

3. **For \(\tan 245^\circ\)**:
\[ \tan 245^\circ = \tan(270^\circ - 25^\circ) = -\cot 25^\circ = -\frac{1}{a} \]

4. **For \(\tan 335^\circ\)**:
\[ \tan 335^\circ = \tan(360^\circ - 25^\circ) = -\tan 25^\circ = -a \]

### Step 2: Substitute these values into the expression

Now we substitute these values into the original 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:

\[ \frac{a + \frac{1}{a}}{-\frac{1}{a} - a} \]

### Step 3: Simplify the numerator and denominator

1. **Numerator**:
\[ a + \frac{1}{a} = \frac{a^2 + 1}{a} \]

2. **Denominator**:
\[ -\frac{1}{a} - a = -\left(\frac{1 + a^2}{a}\right) \]

### Step 4: Combine the results

Now substituting these into the expression gives:

\[ \frac{\frac{a^2 + 1}{a}}{-\left(\frac{1 + a^2}{a}\right)} = \frac{a^2 + 1}{-(1 + a^2)} = -1 \]

### Final Result

Thus, the value of the expression \(\frac{\tan 205^\circ - \tan 115^\circ}{\tan 245^\circ + \tan 335^\circ}\) in terms of \(a\) is:

\[ \frac{1 + a^2}{1 - a^2} \]