If the value of ` theta = 30^(@)` , then value of ` tan^(2) theta + cot ^(2) theta `
A. 1/3
B. 4/3
C. 9/3
D. 10/3

Options

A. D
B. B
C. C
D. A

Answer

To solve the problem, we need to find the value of \( \tan^2 \theta + \cot^2 \theta \) when \( \theta = 30^\circ \).

### Step-by-Step Solution:

1. **Identify the values of \( \tan \theta \) and \( \cot \theta \)**:
- We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \).
- The cotangent is the reciprocal of the tangent, so \( \cot 30^\circ = \sqrt{3} \).

2. **Calculate \( \tan^2 \theta \)**:
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

3. **Calculate \( \cot^2 \theta \)**:
\[ \cot^2 30^\circ = (\sqrt{3})^2 = 3 \]

4. **Add \( \tan^2 \theta \) and \( \cot^2 \theta \)**:
\[ \tan^2 30^\circ + \cot^2 30^\circ = \frac{1}{3} + 3 \]

5. **Convert \( 3 \) into a fraction with a common denominator**:
- Convert \( 3 \) to a fraction:
\[ 3 = \frac{9}{3} \]

6. **Combine the fractions**:
\[ \tan^2 30^\circ + \cot^2 30^\circ = \frac{1}{3} + \frac{9}{3} = \frac{10}{3} \]

7. **Final Result**:
The value of \( \tan^2 \theta + \cot^2 \theta \) is \( \frac{10}{3} \).

### Conclusion:
Thus, the correct answer is option D: \( \frac{10}{3} \).