The value of tan `9^@ - tan 27^@ - tan 63^@ + tan 81^@` is equal to:

Options

A. A) `-1`
B. B) 0
C. C) 1
D. D) 4

Answer

To solve the expression \( \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ \), we can use some properties of trigonometric functions.

### Step-by-Step Solution:

1. **Rewrite the terms**:
We can use the identity \( \tan(90^\circ - \theta) = \cot \theta \) to rewrite \( \tan 81^\circ \) and \( \tan 63^\circ \):
\[ \tan 81^\circ = \cot 9^\circ \quad \text{and} \quad \tan 63^\circ = \cot 27^\circ \]
Therefore, we can rewrite the expression as:
\[ \tan 9^\circ - \tan 27^\circ - \cot 27^\circ + \cot 9^\circ \]

2. **Group the terms**:
Now, we can group the terms:
\[ (\tan 9^\circ + \cot 9^\circ) - (\tan 27^\circ + \cot 27^\circ) \]

3. **Use the identity**:
We know that \( \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \). Thus, we can rewrite the grouped terms:
\[ \tan 9^\circ + \cot 9^\circ = \frac{1}{\sin 9^\circ \cos 9^\circ} = \frac{2}{\sin 18^\circ} \]
and
\[ \tan 27^\circ + \cot 27^\circ = \frac{1}{\sin 27^\circ \cos 27^\circ} = \frac{2}{\sin 54^\circ} \]

4. **Substituting back**:
Substitute these back into the expression:
\[ \frac{2}{\sin 18^\circ} - \frac{2}{\sin 54^\circ} \]

5. **Finding a common denominator**:
The common denominator is \( \sin 18^\circ \sin 54^\circ \):
\[ = \frac{2 \sin 54^\circ - 2 \sin 18^\circ}{\sin 18^\circ \sin 54^\circ} \]

6. **Using the sine subtraction formula**:
We can simplify \( \sin 54^\circ - \sin 18^\circ \) using the sine subtraction formula:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Here, \( A = 54^\circ \) and \( B = 18^\circ \):
\[ = 2 \cos\left(36^\circ\right) \sin\left(18^\circ\right) \]

7. **Substituting back**:
Substitute this back into the expression:
\[ = \frac{2 \cdot 2 \cos(36^\circ) \sin(18^\circ)}{\sin 18^\circ \sin 54^\circ} \]
The \( \sin 18^\circ \) cancels out:
\[ = \frac{4 \cos(36^\circ)}{\sin 54^\circ} \]

8. **Using known values**:
We know that \( \sin 54^\circ = \cos 36^\circ \), thus:
\[ = \frac{4 \cos(36^\circ)}{\cos(36^\circ)} = 4 \]

### Final Answer:
Thus, the value of \( \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ \) is \( \boxed{4} \).