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The given problem asks to prove that:

\[
\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}.
\]

### Proof:

1. Let:
   \[
   \theta_1 = \sin^{-1}(x) \quad \text{and} \quad \theta_2 = \cos^{-1}(x).
   \]
   This means that:
   \[
   \sin(\theta_1) = x \quad \text{and} \quad \cos(\theta_2) = x.
   \]
   
2. By the definition of inverse trigonometric functions, we know that \(\theta_1\) is the angle whose sine is \(x\) and \(\theta_2\) is the angle whose cosine is \(x\). We need to show that:
   \[
   \theta_1 + \theta_2 = \frac{\pi}{2}.
   \]
   
3. From basic trigonometry, we know that for any angle \(\theta\):
   \[
   \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right).
   \]
   
4. If we substitute \(\theta_1\) for \(\theta\), we have:
   \[
   \sin(\theta_1) = \cos\left(\frac{\pi}{2} - \theta_1\right).
   \]
   
   But we are given that \(\sin(\theta_1) = x\). Therefore:
   \[
   x = \cos\left(\frac{\pi}{2} - \theta_1\right).
   \]
   
5. Now compare this with \(\cos(\theta_2) = x\). Since both \(\cos(\theta_2)\) and \(\cos\left(\frac{\pi}{2} - \theta_1\right)\) equal \(x\), we conclude:
   \[
   \theta_2 = \frac{\pi}{2} - \theta_1.
   \]
   
6. Rearranging this gives:
   \[
   \theta_1 + \theta_2 = \frac{\pi}{2}.
   \]

Thus, we have proven that:

\[
\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}.
\]

Would you like more details on any part of the proof?

### Follow-up questions:
1. What is the domain of the inverse sine and cosine functions?
2. How would the proof change if \(x\) were outside the range \([-1, 1]\)?
3. Can we prove similar identities for other inverse trigonometric functions?
4. What is the geometric interpretation of this identity?
5. How do we derive this identity from the unit circle?

### Tip:
For inverse trigonometric functions, always keep the principal values and their domains in mind when working with identities.

Learn how to prove the inverse trigonometric identity sin⁻¹(x) + cos⁻¹(x) = π/2. This detailed step-by-step proof involves understanding inverse sine and cosine functions, trigonometric identities, and basic trigonometry. Suitable for high school students (Grades 11-12).

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