If `cot theta+tan theta=2`, then the value of `tan^(2) theta- cot^(2) theta` is _______

Options

A. 1
B. 0
C. `-1`
D. `2`

Explanation

To solve the equation \( \cot \theta + \tan \theta = 2 \) and find the value of \( \tan^2 \theta - \cot^2 \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We know that:
\[ \tan \theta = \frac{1}{\cot \theta} \]
Thus, we can rewrite the equation:
\[ \cot \theta + \frac{1}{\cot \theta} = 2 \]

### Step 2: Let \( x = \cot \theta \)
Substituting \( x \) for \( \cot \theta \), we get:
\[ x + \frac{1}{x} = 2 \]

### Step 3: Multiply through by \( x \)
To eliminate the fraction, multiply both sides by \( x \):
\[ x^2 + 1 = 2x \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ x^2 - 2x + 1 = 0 \]

### Step 5: Factor the quadratic
This can be factored as:
\[ (x - 1)^2 = 0 \]

### Step 6: Solve for \( x \)
Setting the factor equal to zero gives:
\[ x - 1 = 0 \implies x = 1 \]
Thus, \( \cot \theta = 1 \) which implies \( \tan \theta = 1 \) as well.

### Step 7: Find \( \tan^2 \theta - \cot^2 \theta \)
Now we can substitute back to find \( \tan^2 \theta - \cot^2 \theta \):
\[ \tan^2 \theta - \cot^2 \theta = 1^2 - 1^2 = 1 - 1 = 0 \]

### Final Answer
The value of \( \tan^2 \theta - \cot^2 \theta \) is:
\[ \boxed{0} \]