The value of (secA + tanA-1) (sec A -tanA+1) -2tanA is equal to

Options

A. sec A
B. 2 secA
C. 0
D. 1

Answer

To solve the expression \((\sec A + \tan A - 1)(\sec A - \tan A + 1) - 2\tan A\), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the given expression:
\[ (\sec A + \tan A - 1)(\sec A - \tan A + 1) - 2\tan A \]

### Step 2: Identify the Structure
Notice that the expression can be treated as a product of two binomials. We can rewrite it as:
\[ \text{Let } x = \sec A \text{ and } y = \tan A - 1 \]
Thus, we can express the product as:
\[ (x + y)(x - y) \]
This is in the form of \(A + B\) and \(A - B\), which can be simplified using the identity:
\[ A^2 - B^2 \]

### Step 3: Apply the Identity
Using the identity:
\[ (\sec A)^2 - (\tan A - 1)^2 \]
We can simplify it to:
\[ \sec^2 A - (\tan^2 A - 2\tan A + 1) \]

### Step 4: Expand the Expression
Now, we expand the expression:
\[ \sec^2 A - \tan^2 A + 2\tan A - 1 \]

### Step 5: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 A - \tan^2 A = 1 \]
So we substitute this into our expression:
\[ 1 + 2\tan A - 1 - 2\tan A \]

### Step 6: Simplify the Expression
Now, we simplify the expression:
\[ 1 - 1 + 2\tan A - 2\tan A = 0 \]

### Conclusion
Thus, the value of the original expression is:
\[ \boxed{0} \]