To solve \( \cos^{-1}(\sin(\frac{\pi}{4})) \), follow these steps:

### Step 1: Evaluate \( \sin(\frac{\pi}{4}) \)
We know that:
\[
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]

### Step 2: Now find \( \cos^{-1}\left(\frac{\sqrt{2}}{2}\right) \)
The inverse cosine function gives the angle whose cosine is \( \frac{\sqrt{2}}{2} \).

From trigonometry, we know:
\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]

Therefore:
\[
\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}
\]

### Final Answer:
\[
\cos^{-1}(\sin(\frac{\pi}{4})) = \frac{\pi}{4}
\]

Would you like further clarification or additional details?

Here are some related questions:
1. What is the value of \( \sin^{-1}(\sin(\frac{\pi}{4})) \)?
2. How do inverse trigonometric functions work in different quadrants?
3. What is the value of \( \cos^{-1}(\sin(\frac{\pi}{3})) \)?
4. How does the range of inverse cosine functions affect solutions?
5. What is the derivative of \( \cos^{-1}(x) \)?

**Tip:** Always remember that inverse trigonometric functions return values within specific ranges: for \( \cos^{-1}(x) \), the range is \( [0, \pi] \).

Learn how to solve cos-1(sin(π/4)) step-by-step using trigonometric identities. This problem covers key concepts of inverse trigonometric functions, including evaluating sin(π/4) and finding the inverse cosine. Suitable for students in Grades 10-12.

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