The simplidied value of `(sec x secy +tan x tany)^2-(sec x tan y + tan x secy)^2` is

Options

A. `-1`
B. 0
C. `sec^2x`
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: Identify 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: Apply the difference of squares formula
We can use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\). Here, let:
- \(a = \sec x \sec y + \tan x \tan y\)
- \(b = \sec x \tan y + \tan x \sec y\)

Thus, we can rewrite the expression as:
\[ (a - b)(a + b) \]

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

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

### Step 5: Substitute back into the expression
Now we substitute \(a - b\) and \(a + b\) back into our 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 further
We can simplify the expression further by recognizing that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Using this identity, we can simplify the product of the two terms.

### Final Result
After simplifying, we find that the expression simplifies to:
\[ 1 \]

Thus, the simplified value of \((\sec x \sec y + \tan x \tan y)^2 - (\sec x \tan y + \tan x \sec y)^2\) is:
\[ \boxed{1} \]