What is the simplified value of `(sec^(4)A-tan^(2)A)-(tan^(4)A+sec^(2)A)`?

Options

A. -1
B. `(-1)/(2)`
C. 0
D. 1

Answer

To simplify the expression \( (sec^4 A - tan^2 A) - (tan^4 A + sec^2 A) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ sec^4 A - tan^2 A - (tan^4 A + sec^2 A) \]
This can be rewritten as:
\[ sec^4 A - tan^2 A - tan^4 A - sec^2 A \]

### Step 2: Group the terms
Now, we can group the terms:
\[ (sec^4 A - sec^2 A) - (tan^4 A + tan^2 A) \]

### Step 3: Factor out common terms
From the first group \( sec^4 A - sec^2 A \), we can factor out \( sec^2 A \):
\[ sec^2 A (sec^2 A - 1) \]
Using the identity \( sec^2 A - 1 = tan^2 A \), we can substitute:
\[ sec^2 A \cdot tan^2 A \]

For the second group \( tan^4 A + tan^2 A \), we can factor out \( tan^2 A \):
\[ tan^2 A (tan^2 A + 1) \]
Using the identity \( tan^2 A + 1 = sec^2 A \), we can substitute:
\[ tan^2 A \cdot sec^2 A \]

### Step 4: Combine the factored terms
Now, we can combine the factored terms:
\[ sec^2 A \cdot tan^2 A - tan^2 A \cdot sec^2 A \]
This simplifies to:
\[ 0 \]

### Final Answer
Thus, the simplified value of the original expression is:
\[ \boxed{0} \]