The simplified value of `(secxsecy+tanxtany)^(2)-(secxtany+tanxsecy)^(2)` is :

Options

A. `-1`
B. 0
C. `sec^(2)x`
D. 1

Answer

To simplify the expression \((\sec x \sec y + \tan x \tan y)^2 - (\sec x \tan y + \tan x \sec y)^2\), we can follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ (\sec x \sec y + \tan x \tan y)^2 - (\sec x \tan y + \tan x \sec y)^2 \]

### Step 2: Use the identity for the difference of squares
Recall the identity for the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]
Let \(a = \sec x \sec y + \tan x \tan y\) and \(b = \sec x \tan y + \tan x \sec y\). Thus, we can rewrite the expression as:
\[ (a - b)(a + b) \]

### Step 3: Calculate \(a - b\)
Now, we calculate \(a - b\):
\[ a - b = (\sec x \sec y + \tan x \tan y) - (\sec x \tan y + \tan x \sec y) \]
This simplifies to:
\[ = \sec x \sec y - \sec x \tan y + \tan x \tan y - \tan x \sec y \]
Rearranging gives:
\[ = \sec x (\sec y - \tan y) + \tan x (\tan y - \sec y) \]

### Step 4: Calculate \(a + b\)
Next, we calculate \(a + b\):
\[ a + b = (\sec x \sec y + \tan x \tan y) + (\sec x \tan y + \tan x \sec y) \]
This simplifies to:
\[ = \sec x \sec y + \sec x \tan y + \tan x \tan y + \tan x \sec y \]
Rearranging gives:
\[ = \sec x (\sec y + \tan y) + \tan x (\tan y + \sec y) \]

### Step 5: Substitute back into the expression
Now we substitute \(a - b\) and \(a + b\) back into the expression:
\[ (\sec x (\sec y - \tan y) + \tan x (\tan y - \sec y))(\sec x (\sec y + \tan y) + \tan x (\tan y + \sec y)) \]

### Step 6: Simplify using trigonometric identities
Using the identity \(\tan^2 \theta + 1 = \sec^2 \theta\), we can simplify:
\[ \sec^2 x - \tan^2 x = 1 \]
and
\[ \sec^2 y - \tan^2 y = 1 \]

### Step 7: Final simplification
After substituting the identities and simplifying, we find that the expression simplifies to:
\[ 1 \]

### Conclusion
Thus, the simplified value of the expression is:
\[ \boxed{1} \]