Step 1: Concept Identification: This problem requires evaluating a trigonometric function of an inverse trigonometric function. A common approach involves constructing a right-angled triangle to represent the inverse trigonometric function and then deriving the required trigonometric ratio.
Step 2: Method Outline: Let \(\theta = \tan^{-1}x\), which means \(\tan\theta = x\). The objective is to determine \(\sin\theta\). This relationship can be visualized using a right-angled triangle.
Step 3: Derivation: Given \(\theta = \tan^{-1}x\), we have \(\tan\theta = x\). This can be expressed as \(\frac{x}{1}\). In a right-angled triangle, \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\). Assigning the side opposite to \(\theta\) as \(x\) and the adjacent side as 1, the hypotenuse can be calculated using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = x^2 + 1^2 = 1 + x^2 \]
\[ \text{Hypotenuse} = \sqrt{1 + x^2} \]
To find \(\sin(\tan^{-1}x)\), which is \(\sin\theta\), we use the formula \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\).
\[ \sin\theta = \frac{x}{\sqrt{1+x^2}} \]
Step 4: Conclusion: Consequently, \(\sin(\tan^{-1}x) = \frac{x}{\sqrt{1+x^2}}\).