If `(x -1) (x^(2) + 1) gt 0`, then find the value of `sin((1)/(2) tan^(-1).(2x)/(1 - x^(2)) - tan^(-1) x)`

Explanation

To solve the problem step by step, we start with the inequality and then move on to the trigonometric expression.

### Step 1: Solve the inequality \( (x - 1)(x^2 + 1) > 0 \)

1. **Identify the factors**: The expression consists of two factors: \( (x - 1) \) and \( (x^2 + 1) \).
2. **Analyze \( x^2 + 1 \)**: This factor is always positive for all real \( x \) because \( x^2 \geq 0 \) implies \( x^2 + 1 > 0 \).
3. **Analyze \( x - 1 \)**: This factor is positive when \( x > 1 \) and negative when \( x < 1 \).
4. **Combine the results**: Since \( x^2 + 1 > 0 \) for all \( x \), the inequality \( (x - 1)(x^2 + 1) > 0 \) depends only on \( (x - 1) > 0 \). Therefore, the solution to the inequality is:
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