If `tantheta+(1)/(tantheta)=2` then the value of `tan^(2)theta+(1)/(tan^(2)theta)` is equal to :

Options

A. a.6
B. b.4
C. c.2
D. d.3

Answer

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

2. **Let \( x = \tan \theta \)**:
This allows us to rewrite the equation as:
\[ x + \frac{1}{x} = 2 \]

3. **Rearranging the Equation**:
We can multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = 2x \]

4. **Rearranging to Form a Quadratic Equation**:
Rearranging gives us:
\[ x^2 - 2x + 1 = 0 \]

5. **Factoring the Quadratic**:
This can be factored as:
\[ (x - 1)^2 = 0 \]

6. **Finding the Value of \( x \)**:
Solving this gives:
\[ x - 1 = 0 \implies x = 1 \]
Therefore, we have:
\[ \tan \theta = 1 \]

7. **Finding \( \tan^2 \theta \)**:
Now, we need to find \( \tan^2 \theta \):
\[ \tan^2 \theta = 1^2 = 1 \]

8. **Calculating \( \tan^2 \theta + \frac{1}{\tan^2 \theta} \)**:
Now we can substitute this value into the expression we need to evaluate:
\[ \tan^2 \theta + \frac{1}{\tan^2 \theta} = 1 + \frac{1}{1} = 1 + 1 = 2 \]

9. **Final Answer**:
Thus, the value of \( \tan^2 \theta + \frac{1}{\tan^2 \theta} \) is:
\[ \boxed{2} \]