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In the provided image, the expression appears to involve inverse trigonometric functions, specifically the sine and cosine inverse functions. Here's a breakdown of what I can see:

The expression includes:
1. \(\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\)
2. \(\cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)\)

These expressions often appear in problems that involve the sum or difference of inverse trigonometric functions. The form of \(\frac{2x}{1 + x^2}\) and \(\frac{1 - x^2}{1 + x^2}\) are typical transformations for expressing \(\tan^{-1}\) or combinations thereof.

### Possible Approach to Solve or Simplify
The expressions \(\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\) and \(\cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)\) can often be simplified because they represent forms related to angle addition identities. Specifically:
- \(\frac{2x}{1 + x^2} = \sin(2 \tan^{-1}(x))\)
- \(\frac{1 - x^2}{1 + x^2} = \cos(2 \tan^{-1}(x))\)

Thus:
\[
\sin^{-1} \left( \frac{2x}{1 + x^2} \right) + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) = \frac{\pi}{2}
\]

### Explanation
Since \(\sin^{-1} (x) + \cos^{-1} (x) = \frac{\pi}{2}\) for any argument \(x\) within the domain, this property simplifies the entire expression to \(\frac{\pi}{2}\).

Would you like further details on this solution, or do you have additional questions about specific parts of the problem?

### Related Questions
1. How do you derive the identities for \(\frac{2x}{1 + x^2}\) and \(\frac{1 - x^2}{1 + x^2}\) in terms of trigonometric functions?
2. What is the domain of the expression \(\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\)?
3. Can we use a similar approach to solve expressions involving \(\tan^{-1}\)?
4. How does the identity \(\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}\) work for complex numbers?
5. What other trigonometric identities are helpful in simplifying inverse trigonometric expressions?

**Tip:** When working with inverse trigonometric functions, remember their range constraints, as they define the limits of possible simplifications.

Simplify the expression: sin⁻¹(2x / (1 + x²)) + cos⁻¹((1 - x²) / (1 + x²))

Learn how to simplify the expression sin⁻¹(2x / (1 + x²)) + cos⁻¹((1 - x²) / (1 + x²)) using trigonometric identities. This problem covers key concepts in trigonometry and inverse functions, ideal for students in Grades 11-12.

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