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The image you provided contains a mathematical expression. I'll transcribe and solve it for you.

The equation in the image is:

\[
\sin^{-1} \left(\frac{2x}{1+x^2}\right) = \left\{
\begin{array}{ll}
\cos^{-1} \left(\frac{1-x^2}{1+x^2}\right), & \text{if } -1 \leq x \leq 1 \\
\tan^{-1} \left(\frac{2x}{1-x^2}\right), & \text{if } x > 1 \text{ or } x < -1
\end{array}
\right.
\]

Let’s break this down.

### First Case: \(-1 \leq x \leq 1\)

For \( -1 \leq x \leq 1 \):

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

This equation exploits the identity \( \sin^{-1} y + \cos^{-1} y = \frac{\pi}{2} \). If \( y = \frac{2x}{1+x^2} \), then:

\[
\sin^{-1} y = \sin^{-1} \left(\frac{2x}{1+x^2}\right) \quad \text{and} \quad \cos^{-1} y = \cos^{-1} \left(\frac{1-x^2}{1+x^2}\right)
\]

These two should add up to \( \frac{\pi}{2} \), confirming that the equation holds true in this range.

### Second Case: \( x > 1 \) or \( x < -1 \)

For \( x > 1 \) or \( x < -1 \):

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

In this case, the expression is transformed through the tangent function, applicable when \( x \) is outside the interval \([-1, 1]\).

The expression is solved and verified based on the intervals of \( x \).

Would you like more details, or do you have any questions?

Here are some related questions:

1. What are the geometric interpretations of the arcsine, arccosine, and arctangent functions?
2. How do the ranges of \( x \) affect the validity of trigonometric identities?
3. What are the properties of the inverse trigonometric functions?
4. How does the tangent function behave for large values of \( x \)?
5. Can you explore more complex cases involving trigonometric identities?

**Tip:** When dealing with inverse trigonometric functions, always consider the domain and range constraints to avoid invalid solutions.

Explore the identity involving arcsine, arccosine, and arctangent functions with a detailed explanation. Learn how to apply these identities to solve equations involving trigonometric functions. Suitable for advanced mathematics students.

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