Thank you for the clarification. Since this is a multiple-correct type question, let's analyze the other options to see if any others are also correct.

### Given Expression:
\[
\tan^{-1} \left(\sqrt{\frac{1-x}{1+x}} \right)
\]

### Option A: \(\frac{1}{2} \cos^{-1} x\)
We've already determined that this is correct by using the identity:
\[
\tan^{-1} \left(\sqrt{\frac{1-x}{1+x}} \right) = \frac{1}{2} \cos^{-1} x
\]
So, **Option A** is correct.

### Option B: \(\cos^{-1} \left(\sqrt{\frac{1+x}{2}} \right)\)
We can test this option by analyzing the expression inside the cosine inverse function.

The identity for the half-angle formula is:
\[
\cos \left(\frac{\theta}{2}\right) = \sqrt{\frac{1+\cos \theta}{2}}
\]

If we set \( x = \cos \theta \), then:
\[
\cos^{-1} \left(\sqrt{\frac{1+\cos \theta}{2}} \right) = \cos^{-1} \left(\cos \frac{\theta}{2} \right) = \frac{\theta}{2}
\]
Thus, \( \cos^{-1} \left(\sqrt{\frac{1+x}{2}} \right) = \frac{\cos^{-1} x}{2} \).

Therefore, **Option B** is also correct.

### Option C: \(\sin^{-1} \left(\sqrt{\frac{1-x}{2}} \right)\)
Let's apply the identity for the sine half-angle:
\[
\sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1-\cos \theta}{2}}
\]

So if \( x = \cos \theta \), then:
\[
\sin^{-1} \left(\sqrt{\frac{1-\cos \theta}{2}} \right) = \frac{\theta}{2} = \frac{\cos^{-1} x}{2}
\]
Thus, **Option C** is correct.

### Option D: \(\tan^{-1} x\)
Finally, this option states that the original expression is equal to \( \tan^{-1} x \).

There is no trigonometric identity that directly equates \( \tan^{-1} \left(\sqrt{\frac{1-x}{1+x}} \right) \) with \( \tan^{-1} x \), so **Option D** is **incorrect**.

### Final Answer:
The correct answers are **A, B, and C**.

Would you like to explore any specific option further or ask additional questions? Here are some related questions:

1. How does the sine half-angle identity relate to option C?
2. Why does the inverse cosine function appear in multiple correct options?
3. What is the general strategy for solving inverse trigonometric equations?
4. Could there be other equivalent forms for the given expression?
5. How do different trigonometric identities apply to inverse functions?
6. What is the significance of half-angle formulas in trigonometry?
7. Why is option D incorrect, despite involving an inverse tangent?
8. What are some common pitfalls when solving multiple correct-type questions?

**Tip:** Always test each option thoroughly in multiple-correct questions, as more than one answer may be right due to different equivalent forms of the expression.

Explore how to solve inverse trigonometric equations with multiple correct options. Understand identities like half-angle formulas for cosine and sine and their application in solving complex trigonometric expressions. Learn why specific options are correct or incorrect in trigonometric problem-solving.

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