If `(secA-cosA)(SecA+cosA)=`x , then value of x is

Options

A. `cos^(2)A+cot^(2)A`
B. `sin^(2)A+tan^(2)A`
C. `sin^(2)A-tan^(2)A`
D. `cos^(2)A-cot^(2)A`

Answer

To solve the problem `(secA - cosA)(secA + cosA) = x`, we can follow these steps:

### Step 1: Recognize the Expression
The expression `(secA - cosA)(secA + cosA)` can be recognized as a difference of squares. This follows the algebraic identity:
\[ (A - B)(A + B) = A^2 - B^2 \]
Here, we can let \( A = secA \) and \( B = cosA \).

### Step 2: Apply the Difference of Squares Formula
Using the difference of squares formula, we can rewrite the expression:
\[ x = sec^2A - cos^2A \]

### Step 3: Substitute Trigonometric Identities
We know from trigonometric identities that:
\[ sec^2A = 1 + tan^2A \]
and
\[ cos^2A = 1 - sin^2A \]
Now we substitute these identities into our expression for \( x \):
\[ x = (1 + tan^2A) - cos^2A \]

### Step 4: Substitute for cos^2A
We can also express \( cos^2A \) in terms of \( sin^2A \):
\[ cos^2A = 1 - sin^2A \]
Thus, we can rewrite \( x \) as:
\[ x = (1 + tan^2A) - (1 - sin^2A) \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ x = 1 + tan^2A - 1 + sin^2A \]
This simplifies to:
\[ x = tan^2A + sin^2A \]

### Step 6: Final Expression
Thus, we have the final expression for \( x \):
\[ x = tan^2A + sin^2A \]

### Conclusion
Therefore, the value of \( x \) is:
\[ x = tan^2A + sin^2A \]