The value of `tan^6(15^@)-15tan^4(15^@)+15tan^2(15^@)-3` is

Options

A. `-1`
B. 1
C. 2
D. `-2`

Answer

To find the value of \( \tan^6(15^\circ) - 15\tan^4(15^\circ) + 15\tan^2(15^\circ) - 3 \), we can follow these steps:

### Step 1: Find \( \tan(15^\circ) \)
Using the angle subtraction formula for tangent:
\[ \tan(15^\circ) = \tan(45^\circ - 30^\circ) = \frac{\tan(45^\circ) - \tan(30^\circ)}{1 + \tan(45^\circ) \tan(30^\circ)} \]
Substituting the known values:
\[ \tan(45^\circ) = 1, \quad \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus,
\[ \tan(15^\circ) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]
Now, we can rationalize the denominator:
\[ \tan(15^\circ) = \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{3 - 2\sqrt{3} + 1}{3 - 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]

### Step 2: Substitute \( \tan(15^\circ) \) into the expression
Let \( x = \tan(15^\circ) = 2 - \sqrt{3} \). We need to evaluate:
\[ x^6 - 15x^4 + 15x^2 - 3 \]

### Step 3: Calculate \( x^2 \)
\[ x^2 = (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

### Step 4: Calculate \( x^4 \)
\[ x^4 = (x^2)^2 = (7 - 4\sqrt{3})^2 = 49 - 56\sqrt{3} + 48 = 97 - 56\sqrt{3} \]

### Step 5: Calculate \( x^6 \)
\[ x^6 = x^2 \cdot x^4 = (7 - 4\sqrt{3})(97 - 56\sqrt{3}) \]
Calculating this:
\[ = 679 - 392\sqrt{3} - 388\sqrt{3} + 224 \cdot 3 = 679 + 672 - 780\sqrt{3} = 1351 - 780\sqrt{3} \]

### Step 6: Substitute back into the expression
Now substituting into the expression:
\[ x^6 - 15x^4 + 15x^2 - 3 = (1351 - 780\sqrt{3}) - 15(97 - 56\sqrt{3}) + 15(7 - 4\sqrt{3}) - 3 \]
Calculating \( -15x^4 \):
\[ -15(97 - 56\sqrt{3}) = -1455 + 840\sqrt{3} \]
Calculating \( 15x^2 \):
\[ 15(7 - 4\sqrt{3}) = 105 - 60\sqrt{3} \]
Now combine all terms:
\[ = (1351 - 780\sqrt{3} - 1455 + 840\sqrt{3} + 105 - 60\sqrt{3} - 3) \]
Combine the constants:
\[ 1351 - 1455 + 105 - 3 = -2 \]
Combine the coefficients of \( \sqrt{3} \):
\[ -780 + 840 - 60 = 0 \]
Thus, the final result is:
\[ \boxed{-2} \]