For any real number ` x ge 1`, the expression
`sec^(2) ( tan^(-1)x) - tan^(2) ( sec^(-1) x)` is equal to

Options

A. 1
B. 2
C. `2x^(2)`
D. `2sqrt2`

Answer

To solve the expression \( \sec^2(\tan^{-1} x) - \tan^2(\sec^{-1} x) \) for any real number \( x \geq 1 \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sec^2(\tan^{-1} x) - \tan^2(\sec^{-1} x) \]

### Step 2: Let \( \tan^{-1} x = \alpha \)
Let \( \tan^{-1} x = \alpha \). Then, we have:
\[ x = \tan \alpha \]

### Step 3: Find \( \sec^2(\tan^{-1} x) \)
Using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \):
\[ \sec^2(\tan^{-1} x) = \sec^2 \alpha = 1 + \tan^2 \alpha = 1 + x^2 \]

### Step 4: Let \( \sec^{-1} x = \beta \)
Now, let \( \sec^{-1} x = \beta \). Then, we have:
\[ x = \sec \beta \]

### Step 5: Find \( \tan^2(\sec^{-1} x) \)
Using the identity \( \tan^2 \beta = \sec^2 \beta - 1 \):
\[ \tan^2(\sec^{-1} x) = \tan^2 \beta = \sec^2 \beta - 1 = x^2 - 1 \]

### Step 6: Substitute back into the expression
Now we substitute the values we found back into the original expression:
\[ \sec^2(\tan^{-1} x) - \tan^2(\sec^{-1} x) = (1 + x^2) - (x^2 - 1) \]

### Step 7: Simplify the expression
Simplifying the expression gives us:
\[ 1 + x^2 - x^2 + 1 = 2 \]

### Conclusion
Thus, the value of the expression \( \sec^2(\tan^{-1} x) - \tan^2(\sec^{-1} x) \) is:
\[ \boxed{2} \]