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 $[0, \pi]$.