To simplify the expression \( \sqrt{1 - \sin^2(\sin^{-1}(x))} \), let's go step by step:

### Step 1: Understanding \( \sin^{-1}(x) \)
The term \( \sin^{-1}(x) \) represents the inverse sine (or arcsine) of \( x \), which gives us an angle \( \theta \) such that:
\[
\sin(\theta) = x
\]

Thus, \( \sin^{-1}(x) \) is an angle \( \theta \), and we can replace \( \sin^{-1}(x) \) with \( \theta \):
\[
\theta = \sin^{-1}(x)
\]
This implies \( \sin(\theta) = x \).

### Step 2: Simplifying the expression
Now substitute this back into the expression:
\[
\sqrt{1 - \sin^2(\theta)}
\]

We already know that \( \sin(\theta) = x \), so:
\[
\sqrt{1 - \sin^2(\theta)} = \sqrt{1 - x^2}
\]

### Final Answer:
The simplified form of the expression is:
\[
\sqrt{1 - x^2}
\]

Would you like a further explanation, or do you have any other questions?

### Related Questions:
1. What are the properties of inverse trigonometric functions?
2. How do you derive the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \)?
3. How do you find the derivative of \( \sin^{-1}(x) \)?
4. What is the domain and range of the function \( \sin^{-1}(x) \)?
5. Can you explain how to simplify expressions involving inverse trigonometric functions?

**Tip:** Remember that \( \sin^{-1}(x) \) gives you an angle whose sine is \( x \), and this identity helps simplify many trigonometric expressions.

Discover how to simplify the trigonometric expression √(1 - sin^2(sin^(-1)x)) step-by-step. This solution involves key trigonometric concepts, including inverse sine and the Pythagorean identity, and is suitable for high school students in Grades 10-12.

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