If `x=sectheta-tantheta, y=sectheta+tantheta`, 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 \( y \) given \( x = \sec \theta - \tan \theta \) 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 \( y \)
Now, we will multiply \( x \) and \( y \):
\[ xy = (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) \]

### 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 the expression:
\[ xy = \sec^2 \theta - \tan^2 \theta \]

### 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:
\[ xy = 1 \]

### Step 5: Conclusion
Therefore, the relation between \( x \) and \( y \) is:
\[ xy = 1 \]

### Final Answer
The relation between \( x \) and \( y \) is \( xy = 1 \).

---