`("sec"theta+"tan"theta)/("sec"theta-"tan"theta)` is equal to:

Options

A. A)`1/("sec"theta-"tan"theta)`
B. B)`1/("sec"theta+"tan"theta)`
C. C)`("sec"theta+"tan"theta)^(2)`
D. D)`("sec"theta-"tan"theta)^(2)`

Answer

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

### Step 1: Multiply by the Conjugate
We will multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sec \theta + \tan \theta\).

\[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} \cdot \frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta} \]

### Step 2: Apply the Difference of Squares Formula
This multiplication gives us:

\[ \frac{(\sec \theta + \tan \theta)^2}{(\sec \theta - \tan \theta)(\sec \theta + \tan \theta)} \]

Using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\), we can simplify the denominator:

\[ \sec^2 \theta - \tan^2 \theta \]

### Step 3: Simplify the Denominator
We know from the Pythagorean identity that:

\[ \sec^2 \theta - \tan^2 \theta = 1 \]

So, we can substitute this into our expression:

\[ \frac{(\sec \theta + \tan \theta)^2}{1} = (\sec \theta + \tan \theta)^2 \]

### Final Answer
Thus, the expression simplifies to:

\[ \sec \theta + \tan \theta)^2 \]