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

Options

A. `(pi)/(4)-(1)/(2) cos^(-1)x`
B. `(pi)/(4) +cos^(-1)x^(2) `
C. `(pi)/(4) +(1)/(2) cos^(-1)x^(2)`
D. `(pi)/(4) - (1)/(2) cos^(-1)x^(2)`

Answer

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

### Step 1: Substitute \( x^2 \)
Let \( x^2 = \cos(2\theta) \). Then, we can express \( \sqrt{1+x^2} \) and \( \sqrt{1-x^2} \) in terms of \( \theta \).

### Step 2: Simplify \( \sqrt{1+x^2} \)
Using the identity \( 1 + \cos(2\theta) = 2\cos^2(\theta) \):
\[ \sqrt{1+x^2} = \sqrt{1 + \cos(2\theta)} = \sqrt{2\cos^2(\theta)} = \sqrt{2} \cos(\theta) \]

### Step 3: Simplify \( \sqrt{1-x^2} \)
Using the identity \( 1 - \cos(2\theta) = 2\sin^2(\theta) \):
\[ \sqrt{1-x^2} = \sqrt{1 - \cos(2\theta)} = \sqrt{2\sin^2(\theta)} = \sqrt{2} \sin(\theta) \]

### Step 4: Substitute back into the expression
Now, substitute these values back into the original expression:
\[ \tan^{-1}\left(\frac{\sqrt{2}\cos(\theta) + \sqrt{2}\sin(\theta)}{\sqrt{2}\cos(\theta) - \sqrt{2}\sin(\theta)}\right) \]

### Step 5: Factor out \( \sqrt{2} \)
Factoring out \( \sqrt{2} \) from the numerator and the denominator:
\[ = \tan^{-1}\left(\frac{\sqrt{2}(\cos(\theta) + \sin(\theta))}{\sqrt{2}(\cos(\theta) - \sin(\theta))}\right) \]
This simplifies to:
\[ = \tan^{-1}\left(\frac{\cos(\theta) + \sin(\theta)}{\cos(\theta) - \sin(\theta)}\right) \]

### Step 6: Use the tangent addition formula
We can rewrite this as:
\[ = \tan^{-1\left(\frac{1 + \tan(\theta)}{1 - \tan(\theta)}\right) \]
This is equivalent to \( \tan\left(\frac{\pi}{4} + \theta\right) \).

### Step 7: Solve for \( \theta \)
Since \( x^2 = \cos(2\theta) \), we have:
\[ 2\theta = \cos^{-1}(x) \quad \Rightarrow \quad \theta = \frac{1}{2} \cos^{-1}(x) \]

### Step 8: Substitute back to find the final answer
Substituting \( \theta \) back, we get:
\[ \tan^{-1}\left(\frac{\cos(\theta) + \sin(\theta)}{\cos(\theta) - \sin(\theta)}\right) = \frac{\pi}{4} + \frac{1}{2} \cos^{-1}(x) \]

Thus, the final answer is:
\[ \frac{\pi}{4} + \frac{1}{2} \cos^{-1}(x) \]

### Conclusion
The expression simplifies to:
\[ \frac{\pi}{4} + \frac{1}{2} \cos^{-1}(x) \]