If `x=tan15^(@)`, then find the value of `x^(2)+(1)/(x^(2))`.

Options

A. 16
B. 15
C. 14
D. 18

Answer

To solve the problem where \( x = \tan(15^\circ) \) and we need to find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the value of \( x \)**:
\[ x = \tan(15^\circ) \]
We know from trigonometric tables or calculations that:
\[ \tan(15^\circ) = 2 - \sqrt{3} \]

2. **Calculate \( x^2 \)**:
\[ x^2 = (2 - \sqrt{3})^2 \]
Expanding this using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ x^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
\[ = 4 - 4\sqrt{3} + 3 \]
\[ = 7 - 4\sqrt{3} \]

3. **Calculate \( \frac{1}{x} \)**:
Using the identity \( \frac{1}{\tan(\theta)} = \cot(\theta) \):
\[ \frac{1}{x} = \cot(15^\circ) \]
We know that:
\[ \cot(15^\circ) = \tan(75^\circ) = 2 + \sqrt{3} \]

4. **Calculate \( \frac{1}{x^2} \)**:
\[ \frac{1}{x^2} = (2 + \sqrt{3})^2 \]
Expanding this:
\[ = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
\[ = 4 + 4\sqrt{3} + 3 \]
\[ = 7 + 4\sqrt{3} \]

5. **Add \( x^2 \) and \( \frac{1}{x^2} \)**:
\[ x^2 + \frac{1}{x^2} = (7 - 4\sqrt{3}) + (7 + 4\sqrt{3}) \]
The \( -4\sqrt{3} \) and \( +4\sqrt{3} \) cancel out:
\[ = 7 + 7 = 14 \]

### Final Answer:
\[ x^2 + \frac{1}{x^2} = 14 \]