The simplest form of   `(1)/(2) cos^(-1) ((1-x)/(1+x)), x in [0,1]` is

Options

A. `tan^(-1) sqrt(x)`
B. `sin^(-1)x`
C. `cos^(-1) sqrt(x)`
D. `sec^(-1) x`

Answer

To find the simplest form of the expression \(\frac{1}{2} \cos^{-1} \left( \frac{1-x}{1+x} \right)\) for \(x \in [0,1]\), we can follow these steps:

### Step-by-Step Solution:

1. **Recognize the Expression**:
We start with the expression:
\[ y = \frac{1}{2} \cos^{-1} \left( \frac{1-x}{1+x} \right) \]

2. **Use a Trigonometric Identity**:
We know from trigonometric identities that:
\[ \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos(2\theta) \]
If we let \(x = \tan^2 \theta\), then:
\[ \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos(2\theta) \]

3. **Substituting \(x\)**:
Substitute \(x = \tan^2 \theta\) into our expression:
\[ y = \frac{1}{2} \cos^{-1} \left( \cos(2\theta) \right) \]

4. **Simplifying the Inverse Cosine**:
The property of inverse cosine states that:
\[ \cos^{-1}(\cos A) = A \quad \text{for } A \in [0, \pi] \]
Thus:
\[ y = \frac{1}{2} \cdot 2\theta = \theta \]

5. **Finding \(\theta\)**:
Since we defined \(\theta\) as:
\[ \theta = \tan^{-1}(\sqrt{x}) \]
Therefore, we have:
\[ y = \tan^{-1}(\sqrt{x}) \]

6. **Final Result**:
Thus, the simplest form of the expression \(\frac{1}{2} \cos^{-1} \left( \frac{1-x}{1+x} \right)\) is:
\[ \tan^{-1}(\sqrt{x}) \]

### Conclusion:
The simplest form of \(\frac{1}{2} \cos^{-1} \left( \frac{1-x}{1+x} \right)\) is \(\tan^{-1}(\sqrt{x})\).

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