To simplify \( \tan(\sin^{-1}(x)) \), follow these steps:

1. Let \( \theta = \sin^{-1}(x) \), which implies that \( \sin(\theta) = x \).
   - Since \( \theta = \sin^{-1}(x) \), it means \( \theta \) is an angle whose sine is \( x \), or \( \sin(\theta) = x \).
   - This gives us the identity \( \sin(\theta) = \frac{x}{1} \), meaning the opposite side is \( x \) and the hypotenuse is \( 1 \) in a right triangle.

2. Use the Pythagorean theorem to find the adjacent side.
   - The Pythagorean theorem states \( \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \).
   - Therefore, \( 1^2 = x^2 + \text{adjacent}^2 \), so \( \text{adjacent} = \sqrt{1 - x^2} \).

3. Now, the tangent of \( \theta \) is:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{\sqrt{1 - x^2}}.
   \]

Thus, the simplified form of \( \tan(\sin^{-1}(x)) \) is:
\[
\tan(\sin^{-1}(x)) = \frac{x}{\sqrt{1 - x^2}}.
\]

Would you like further details or have any questions?

Here are 5 related questions to expand this:

1. How do you simplify \( \tan(\cos^{-1}(x)) \)?
2. What is the domain of \( \tan(\sin^{-1}(x)) \)?
3. Can you find \( \sin(\tan^{-1}(x)) \)?
4. What are the geometric interpretations of inverse trigonometric functions?
5. How do you simplify \( \tan(\sec^{-1}(x)) \)?

**Tip:** Understanding the triangle relationships when dealing with inverse trigonometric functions helps in simplifying expressions quickly.

Learn how to simplify tan(sin⁻¹(x)) using trigonometric identities and the Pythagorean theorem. This solution breaks down the process step-by-step, making it easy to understand for students in Grades 10-12.

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