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The image contains three expressions that ask you to prove the following identities:

1. \( \cos^{-1}(x) = \sqrt{1 - x^2} \)
2. \( \sin^{-1}(x) = \sqrt{1 - x^2} \)
3. \( \tan^{-1}(x) = \sqrt{1 + x^2} \)

### Let's analyze each identity:

#### 1. Proof of \( \cos^{-1}(x) = \sqrt{1 - x^2} \)

This expression appears incorrect as written. The correct relationship involving \( \cos^{-1}(x) \) (also known as the inverse cosine or arccosine function) doesn't match the form \( \sqrt{1 - x^2} \). Usually, \( \cos^{-1}(x) \) is related to angles, and this particular form resembles the Pythagorean identity for sine. Instead, we have:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
so that \( \sin(\cos^{-1}(x)) = \sqrt{1 - x^2} \), but this would not directly equate to \( \cos^{-1}(x) \) itself.

#### 2. Proof of \( \sin^{-1}(x) = \sqrt{1 - x^2} \)

Similar to the first one, this seems incorrect. However, it's true that \( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \). The inverse sine, or arcsine, is related to the cosine function using the Pythagorean identity. For \( \theta = \sin^{-1}(x) \), we have:
\[
\cos(\theta) = \sqrt{1 - x^2}
\]

Thus, the identity would hold if it were expressed as \( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \).

#### 3. Proof of \( \tan^{-1}(x) = \sqrt{1 + x^2} \)

This form again is not correct for the inverse tangent function. The correct form would likely involve an expression for \( \sin(\tan^{-1}(x)) \) or \( \cos(\tan^{-1}(x)) \), but the square root of \( 1 + x^2 \) is typically seen in the context of secant, such as:
\[
\sec(\tan^{-1}(x)) = \sqrt{1 + x^2}
\]
This arises from the Pythagorean identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

### Summary
The expressions as given seem to be miswritten or confused with different trigonometric identities. I can provide the correct versions of these proofs or help derive related identities if you want.

Would you like further clarifications, or should I guide you through specific derivations of the correct forms?

### Follow-up Questions:
1. Would you like a detailed derivation of the correct identities for each function?
2. Are you looking for inverse trigonometric identities involving sine, cosine, or tangent?
3. Should we focus on understanding the geometric meaning behind the inverse trigonometric functions?
4. Do you want to explore how these identities relate to right-angled triangles?
5. Would you like to review the Pythagorean identities and their applications?

### Tip:
Understanding inverse trigonometric functions often involves recognizing their geometric representations on the unit circle or right triangles.

Prove the following trigonometric identities:
1. cos^(-1)(x) = √(1 - x^2)
2. sin^(-1)(x) = √(1 - x^2)
3. tan^(-1)(x) = √(1 + x^2)

Explore the proofs of inverse trigonometric identities involving cos^(-1)(x), sin^(-1)(x), and tan^(-1)(x). This page covers the key Pythagorean identities and their relationships with inverse functions. Suitable for advanced high school and college students.

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