`(sectheta+tantheta-1)(sectheta-tantheta+1)=`

Options

A. `2 sin theta`
B. `2 cos theta`
C. `2 sec theta`
D. `2 tan theta`

Answer

To solve the expression \((\sec \theta + \tan \theta - 1)(\sec \theta - \tan \theta + 1)\), we can follow these steps:

### Step 1: Identify the expression
We have the expression:
\[ (\sec \theta + \tan \theta - 1)(\sec \theta - \tan \theta + 1) \]

### Step 2: Use the difference of squares formula
We can recognize that this expression can be simplified using the difference of squares formula:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, let \(a = \sec \theta\) and \(b = \tan \theta - 1\). Thus, we can rewrite the expression as:
\[ \sec \theta^2 - (\tan \theta - 1)^2 \]

### Step 3: Expand the square
Next, we need to expand \((\tan \theta - 1)^2\):
\[ (\tan \theta - 1)^2 = \tan^2 \theta - 2\tan \theta + 1 \]

### Step 4: Substitute back into the expression
Now substitute this back into our expression:
\[ \sec^2 \theta - (\tan^2 \theta - 2\tan \theta + 1) \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ \sec^2 \theta - \tan^2 \theta + 2\tan \theta - 1 \]

### Step 6: Use the Pythagorean identity
Recall the Pythagorean identity:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Substituting this identity into our expression gives:
\[ (1 + \tan^2 \theta) - \tan^2 \theta + 2\tan \theta - 1 \]

### Step 7: Final simplification
Now, simplify further:
\[ 1 + \tan^2 \theta - \tan^2 \theta + 2\tan \theta - 1 = 2\tan \theta \]

### Final Answer
Thus, the value of the expression \((\sec \theta + \tan \theta - 1)(\sec \theta - \tan \theta + 1)\) is:
\[ \boxed{2\tan \theta} \]