The value of `(sec^2theta+2tan theta cot theta -tan^2theta)` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the expression \( \sec^2 \theta + 2 \tan \theta \cot \theta - \tan^2 \theta \), we will follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
We will substitute \( \sec^2 \theta \) in the expression.

### Step 2: Substitute in the Expression
Replacing \( \sec^2 \theta \) in the expression gives:
\[ 1 + \tan^2 \theta + 2 \tan \theta \cot \theta - \tan^2 \theta \]

### Step 3: Simplify the Expression
Now, we can simplify the expression:
\[ 1 + \tan^2 \theta - \tan^2 \theta + 2 \tan \theta \cot \theta \]
The \( \tan^2 \theta \) terms cancel each other out:
\[ 1 + 2 \tan \theta \cot \theta \]

### Step 4: Simplify \( 2 \tan \theta \cot \theta \)
Recall that \( \cot \theta = \frac{1}{\tan \theta} \), so:
\[ 2 \tan \theta \cot \theta = 2 \tan \theta \cdot \frac{1}{\tan \theta} = 2 \]

### Step 5: Final Calculation
Now substitute back into the expression:
\[ 1 + 2 = 3 \]

### Final Answer
Thus, the value of \( \sec^2 \theta + 2 \tan \theta \cot \theta - \tan^2 \theta \) is:
\[ \boxed{3} \]