Given , `0lexle(1)/(2)`, then the value of `tan["sin"^(-1){(x)/(sqrt(2))+(sqrt(1-x^(2)))/(sqrt(2))}-sin^(-1)x]` is

Options

A. 1
B. `sqrt(3)`
C. `-1`
D. `(1)/(sqrt(3))`

Explanation

To solve the problem step by step, we need to evaluate the expression:

\[ \tan\left[\sin^{-1}\left(\frac{x}{\sqrt{2}} + \frac{\sqrt{1-x^2}}{\sqrt{2}}\right) - \sin^{-1}(x)\right] \]

Given that \(0 \leq x \leq \frac{1}{2}\).

### Step 1: Rewrite the expression inside the tangent
We can combine the terms inside the sine inverse by taking a common denominator:

\[ \sin^{-1}\left(\frac{x + \sqrt{1-x^2}}{\sqrt{2}}\right) - \sin^{-1}(x) \]

### Step 2: Substitute \( \sin^{-1}(x) = \theta \)
Let \( \theta = \sin^{-1}(x) \). Then, we have:

\[ x = \sin(\theta) \]

Now, substitute \( x \) in the expression:

\[ \tan\left[\sin^{-1}\left(\frac{\sin(\theta) + \sqrt{1 - \sin^2(\theta)}}{\sqrt{2}}\right) - \theta\right] \]

### Step 3: Simplify the square root term
Using the Pythagorean identity, we know that:

\[ \sqrt{1 - \sin^2(\theta)} = \cos(\theta) \]

Thus, the expression becomes:

\[ \tan\left[\sin^{-1}\left(\frac{\sin(\theta) + \cos(\theta)}{\sqrt{2}}\right) - \theta\right] \]

### Step 4: Factor out \( \frac{1}{\sqrt{2}} \)
Now, we can factor out \( \frac{1}{\sqrt{2}} \):

\[ \tan\left[\sin^{-1}\left(\frac{1}{\sqrt{2}}(\sin(\theta) + \cos(\theta))\right) - \theta\right] \]

### Step 5: Recognize the sine addition formula
The term \( \sin(\theta) + \cos(\theta) \) can be expressed using the sine addition formula:

\[ \sin(\theta + \frac{\pi}{4}) = \frac{1}{\sqrt{2}}(\sin(\theta) + \cos(\theta)) \]

Thus, we rewrite the expression as:

\[ \tan\left[\sin^{-1}\left(\sin\left(\theta + \frac{\pi}{4}\right)\right) - \theta\right] \]

### Step 6: Apply the sine inverse property
Using the property of sine inverse, we have:

\[ \tan\left[\theta + \frac{\pi}{4} - \theta\right] = \tan\left(\frac{\pi}{4}\right) \]

### Step 7: Evaluate the tangent
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we conclude that:

\[ \tan\left[\sin^{-1}\left(\frac{x}{\sqrt{2}} + \frac{\sqrt{1-x^2}}{\sqrt{2}}\right) - \sin^{-1}(x)\right] = 1 \]

### Final Answer
Thus, the value of the given expression is:

\[ \boxed{1} \]