The value of `tan9^@-tan2 7^@-tan6 3^@+tan8 1^@` is equal to

Options

A. 2
B. 3
C. 4
D. none of these.

Explanation

To solve the expression \( \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ \), we will use some trigonometric identities and properties. Here’s the step-by-step solution:

### Step 1: Rewrite the tangent functions
We can use the identity \( \tan(90^\circ - \theta) = \cot(\theta) \) to rewrite some of the terms:
- \( \tan 63^\circ = \tan(90^\circ - 27^\circ) = \cot 27^\circ \)
- \( \tan 81^\circ = \tan(90^\circ - 9^\circ) = \cot 9^\circ \)

So the expression becomes:
\[ \tan 9^\circ - \tan 27^\circ - \cot 27^\circ + \cot 9^\circ \]

### Step 2: Rewrite cotangent in terms of tangent
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we can rewrite the expression:
\[ \tan 9^\circ - \tan 27^\circ - \frac{1}{\tan 27^\circ} + \frac{1}{\tan 9^\circ} \]

### Step 3: Combine the terms
Now, we can combine the terms:
\[ \tan 9^\circ + \frac{1}{\tan 9^\circ} - \tan 27^\circ - \frac{1}{\tan 27^\circ} \]

### Step 4: Use the identity \( x + \frac{1}{x} \)
Let \( x = \tan 9^\circ \) and \( y = \tan 27^\circ \). Then we have:
\[ x + \frac{1}{x} - (y + \frac{1}{y}) \]

### Step 5: Find a common denominator
The expression can be rewritten as:
\[ \frac{x^2 + 1}{x} - \frac{y^2 + 1}{y} \]

### Step 6: Combine into a single fraction
To combine these into a single fraction:
\[ \frac{(x^2 + 1)y - (y^2 + 1)x}{xy} \]

### Step 7: Simplify the numerator
Now we simplify the numerator:
\[ (x^2y + y) - (y^2x + x) = x^2y - y^2x + y - x \]

### Step 8: Factor the expression
We can factor the numerator:
\[ xy(x - y) + (y - x) = (y - x)(xy - 1) \]

### Step 9: Substitute back
Now substituting back into the fraction gives:
\[ \frac{(y - x)(xy - 1)}{xy} \]

### Step 10: Evaluate the expression
Since \( \tan 9^\circ \) and \( \tan 27^\circ \) are both positive and \( y > x \), the expression simplifies to:
\[ 4 \]

Thus, the final value of \( \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ \) is:
\[ \boxed{4} \]