What is the simplified value of (sec A + cos A) (sec A - cos A) ?

Options

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

Answer

To simplify the expression \((\sec A + \cos A)(\sec A - \cos A)\), we can follow these steps:

### Step 1: Recognize the form of the expression
The expression \((\sec A + \cos A)(\sec A - \cos A)\) is in the form of \((a + b)(a - b)\), which can be simplified using the difference of squares formula:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, let \(a = \sec A\) and \(b = \cos A\).

### Step 2: Apply the difference of squares
Using the difference of squares formula, we can rewrite the expression:
\[ (\sec A + \cos A)(\sec A - \cos A) = \sec^2 A - \cos^2 A \]

### Step 3: Use trigonometric identities
Now we will use the trigonometric identities:
1. \(\sec^2 A = 1 + \tan^2 A\)
2. \(\cos^2 A = 1 - \sin^2 A\)

### Step 4: Substitute the identities into the expression
Substituting the identities into our expression gives:
\[ \sec^2 A - \cos^2 A = (1 + \tan^2 A) - (1 - \sin^2 A) \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ (1 + \tan^2 A) - (1 - \sin^2 A) = 1 + \tan^2 A - 1 + \sin^2 A \]
The \(1\) and \(-1\) cancel out:
\[ \tan^2 A + \sin^2 A \]

### Final Answer
Thus, the simplified value of \((\sec A + \cos A)(\sec A - \cos A)\) is:
\[ \tan^2 A + \sin^2 A \]