`sec^(-1)((1+tan^(2)x)/(1-tan^(2)x))`

Answer

To solve the expression \( \sec^{-1}\left(\frac{1 + \tan^2 x}{1 - \tan^2 x}\right) \), we will follow these steps:

### Step 1: Use the identity for \( \tan^2 x \)
We know that \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \). Therefore, we can rewrite the expression as:
\[ \sec^{-1}\left(\frac{1 + \frac{\sin^2 x}{\cos^2 x}}{1 - \frac{\sin^2 x}{\cos^2 x}}\right) \]

### Step 2: Simplify the expression
Now, we can simplify the fraction:
\[ \sec^{-1}\left(\frac{\frac{\cos^2 x + \sin^2 x}{\cos^2 x}}{\frac{\cos^2 x - \sin^2 x}{\cos^2 x}}\right) \]
This simplifies to:
\[ \sec^{-1}\left(\frac{\cos^2 x + \sin^2 x}{\cos^2 x - \sin^2 x}\right) \]

### Step 3: Apply Pythagorean identity
Using the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \), we have:
\[ \sec^{-1}\left(\frac{1}{\cos^2 x - \sin^2 x}\right) \]

### Step 4: Use the double angle formula
We know that \( \cos^2 x - \sin^2 x = \cos 2x \). Therefore, we can rewrite the expression as:
\[ \sec^{-1}\left(\frac{1}{\cos 2x}\right) \]

### Step 5: Rewrite in terms of secant
Since \( \sec 2x = \frac{1}{\cos 2x} \), we have:
\[ \sec^{-1}(\sec 2x) \]

### Step 6: Apply the property of inverse secant
Using the property \( \sec^{-1}(\sec \theta) = \theta \) for \( \theta \) in the appropriate range, we find:
\[ \sec^{-1}(\sec 2x) = 2x \]

### Final Answer
Thus, the final value is:
\[ \boxed{2x} \]

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