The value of `tan 9^(@) - tan 27^(@)- tan 63^(@)+ tan 81^(@)` is

Options

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

Answer

To solve the problem \( \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ \), we can use the properties of the tangent function and some trigonometric identities. Here’s a step-by-step solution:

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

### Step 2: Substitute cotangent
Now substitute the cotangent terms:
\[ \tan 9^\circ + \cot 9^\circ - \tan 27^\circ - \cot 27^\circ \]

### Step 3: Use the identity for tangent and cotangent
Recall that \( \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \):
\[ \tan 9^\circ + \cot 9^\circ = \frac{1}{\sin 9^\circ \cos 9^\circ} = \frac{2}{\sin 18^\circ} \]
\[ \tan 27^\circ + \cot 27^\circ = \frac{1}{\sin 27^\circ \cos 27^\circ} = \frac{2}{\sin 54^\circ} \]

### Step 4: Substitute back into the expression
Now substitute these back into the expression:
\[ \frac{2}{\sin 18^\circ} - \frac{2}{\sin 54^\circ} \]

### Step 5: Simplify the expression
Factor out the 2:
\[ 2 \left( \frac{1}{\sin 18^\circ} - \frac{1}{\sin 54^\circ} \right) \]

### Step 6: Use the sine identity
Using the identity \( \sin(54^\circ) = \cos(36^\circ) \) and \( \sin(18^\circ) = \cos(72^\circ) \):
\[ \sin 54^\circ = \cos 36^\circ = \sin(90^\circ - 36^\circ) = \sin 54^\circ \]
Thus, we can rewrite the expression:
\[ \frac{1}{\sin 18^\circ} - \frac{1}{\sin 54^\circ} = \frac{\sin 54^\circ - \sin 18^\circ}{\sin 18^\circ \sin 54^\circ} \]

### Step 7: Calculate the final value
Using known values:
\[ \sin 18^\circ = \frac{\sqrt{5}-1}{4}, \quad \sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos 36^\circ = \frac{\sqrt{5}+1}{4} \]
Substituting these values, we can simplify:
\[ 2 \left( \frac{\frac{\sqrt{5}+1}{4} - \frac{\sqrt{5}-1}{4}}{\frac{\sqrt{5}-1}{4} \cdot \frac{\sqrt{5}+1}{4}} \right) \]
This leads us to:
\[ 2 \left( \frac{2}{\frac{(\sqrt{5}-1)(\sqrt{5}+1)}{16}} \right) = 4 \]

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