If `|cos^-1((1-x^2)/(1+x^2))| < pi/3,` then x belongs to the interval

Options

A. `[-1//sqrt(3),1//sqrt(3)]`
B. `(-1//sqrt(3),1//sqrt(3))`
C. `(0,1//sqrt(3))`
D. none of these

Explanation

To solve the inequality \( |\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)| < \frac{\pi}{3} \), we will follow these steps:

### Step 1: Remove the absolute value
Since the range of \(\cos^{-1}(y)\) is \([0, \pi]\), we can remove the absolute value and rewrite the inequality as:
\[ \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) < \frac{\pi}{3} \]

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