Step 1: Concept Review: This problem requires evaluating a trigonometric function applied to an inverse trigonometric function. The standard approach involves constructing a right-angled triangle to represent the inverse trigonometric function, from which the desired trigonometric ratio can be derived.
Step 2: Core Methodology: Let \( \theta = \tan^{-1}x \). This definition implies that \( \tan\theta = x \). The objective is to determine the value of \( \sin\theta \). A right-angled triangle is an effective visual tool for this relationship.
Step 3: In-depth Analysis: Given \( \theta = \tan^{-1}x \), it follows that \( \tan\theta = x \). This can be expressed as \( \tan\theta = \frac{x}{1} \). In the context of a right-angled triangle, \( \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \). Consequently, we can assign the side opposite to angle \( \theta \) a length of \( x \) and the adjacent side a length of 1. The hypotenuse is calculated using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
\[ \text{Hypotenuse}^2 = x^2 + 1^2 = 1 + x^2 \]
\[ \text{Hypotenuse} = \sqrt{1 + x^2} \]
The task is to find \( \sin(\tan^{-1}x) \), which is equivalent to \( \sin\theta \). The formula for \( \sin\theta \) is \( \frac{\text{Opposite}}{\text{Hypotenuse}} \).
\[ \sin\theta = \frac{x}{\sqrt{1+x^2}} \]
Step 4: Conclusion: Thus, the expression \( \sin(\tan^{-1}x) \) simplifies to \( \frac{x}{\sqrt{1+x^2}} \).