If `x=sec theta+tan theta, y = sec theta-tan theta`, then the relation between x and y is :

Options

A. `x^(2)+y^(2)=0`
B. `x^(2)=y^(2)`
C. `x^(2)=y`
D. `xy=1`

Answer

To find the relation between \( x \) and \( y \) where \( x = \sec \theta + \tan \theta \) and \( y = \sec \theta - \tan \theta \), we can follow these steps:

### Step 1: Write down the expressions for \( x \) and \( y \)
We have:
\[ x = \sec \theta + \tan \theta \]
\[ y = \sec \theta - \tan \theta \]

### Step 2: Multiply \( x \) and \( y \)
Next, we multiply \( x \) and \( y \):
\[ x \cdot y = (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) \]

### Step 3: Apply the difference of squares formula
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \), we can simplify:
\[ x \cdot y = \sec^2 \theta - \tan^2 \theta \]

### Step 4: Use the Pythagorean identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Thus, we can substitute this into our equation:
\[ x \cdot y = 1 \]

### Conclusion
The relation between \( x \) and \( y \) is:
\[ x \cdot y = 1 \]