If `(sec^(4)alpha)/(sec^(2)beta)-(tan^(4)alpha)/(tan^(2)beta)=1` where `alpha,betane(pi)/(2)`, then find the value of `(sec^(4)beta)/(sec^(2)alpha)-(tan^(4)beta)/(tan^(2)alpha)`

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the equation

\[ \frac{\sec^4 \alpha}{\sec^2 \beta} - \frac{\tan^4 \alpha}{\tan^2 \beta} = 1 \]

where \(\alpha, \beta \neq \frac{\pi}{2}\), we want to find the value of

\[ \frac{\sec^4 \beta}{\sec^2 \alpha} - \frac{\tan^4 \beta}{\tan^2 \alpha}. \]

### Step 1: Use Trigonometric Identities
Recall the identity:

\[ \sec^2 \theta = 1 + \tan^2 \theta. \]

From this, we can derive that:

\[ \sec^2 \theta - \tan^2 \theta = 1. \]

### Step 2: Substitute \(\beta = \alpha\)
To simplify the original equation, let's substitute \(\beta = \alpha\):

\[ \frac{\sec^4 \alpha}{\sec^2 \alpha} - \frac{\tan^4 \alpha}{\tan^2 \alpha} = 1. \]

This simplifies to:

\[ \sec^2 \alpha - \tan^2 \alpha = 1. \]

### Step 3: Verify the Identity
Since we know that \(\sec^2 \alpha - \tan^2 \alpha = 1\) is true, the original equation holds.

### Step 4: Find the New Expression
Now, we need to evaluate:

\[ \frac{\sec^4 \beta}{\sec^2 \alpha} - \frac{\tan^4 \beta}{\tan^2 \alpha}. \]

By substituting \(\beta = \alpha\):

\[ \frac{\sec^4 \alpha}{\sec^2 \alpha} - \frac{\tan^4 \alpha}{\tan^2 \alpha} = \sec^2 \alpha - \tan^2 \alpha. \]

### Step 5: Substitute the Identity Again
Using the identity again:

\[ \sec^2 \alpha - \tan^2 \alpha = 1. \]

### Conclusion
Thus, the value of

\[ \frac{\sec^4 \beta}{\sec^2 \alpha} - \frac{\tan^4 \beta}{\tan^2 \alpha} = 1. \]

### Final Answer
The final answer is:

\[ \boxed{1}. \]