The complete set of values of x for which `2 tan^(-1)x+cos^(-1)((1-x^(2))/(1+x^(2)))` is independent of x is :

Options

A. `(-oo, 0]`
B. `[0,oo)`
C. `(-oo, -1]`
D. `[1, oo)`

Answer

To solve the problem, we need to find the complete set of values of \( x \) for which the expression

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

is independent of \( x \).

### Step 1: Rewrite the expression

We start with the expression:

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

### Step 2: Use the identity for cosine

We know that:

\[ \cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)} \]

Let \( x = \tan(\theta) \). Then, \( \tan^{-1}(x) = \theta \), and we can rewrite the expression as:

\[ y = 2\theta + \cos^{-1}(\cos(2\theta)) \]

### Step 3: Simplify the expression

Since \( \cos^{-1}(\cos(2\theta)) \) can be simplified to \( 2\theta \) when \( 2\theta \) is within the range of \( \cos^{-1} \), we have:

\[ y = 2\theta + 2\theta = 4\theta \]

### Step 4: Determine when \( y \) is independent of \( x \)

The expression \( 4\theta \) is independent of \( x \) if \( \theta \) itself is constant. Since \( \theta = \tan^{-1}(x) \), for \( \theta \) to be constant, \( x \) must be constant.

### Step 5: Find the values of \( x \)

The only way \( \tan^{-1}(x) \) can be constant is if \( x \) is a specific value. The simplest case is when \( x = 0 \).

Thus, the expression is independent of \( x \) when \( x = 0 \).

### Step 6: Complete set of values

The complete set of values of \( x \) for which the expression is independent of \( x \) is:

\[ \boxed{(-\infty, 0)} \]