Step 1: Use the identity connecting tan and sec:
Let $\theta=\tan^{-1}x$, so $\tan\theta=x$. Use the identity $\sec^2\theta=1+\tan^2\theta$ instead of drawing a triangle.
Step 2: Find cos theta algebraically:
Substitute $\tan\theta=x$ into the identity.
\[ \sec^2\theta = 1+x^2 \quad\Rightarrow\quad \cos\theta=\dfrac{1}{\sqrt{1+x^2}} \]
The positive root is taken since $\theta\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ where cosine stays positive.
Step 3: Find sin theta from tan and cos:
Use $\sin\theta=\tan\theta\cdot\cos\theta$, since $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$.
\[ \sin\theta = x\cdot\dfrac{1}{\sqrt{1+x^2}} = \dfrac{x}{\sqrt{1+x^2}} \]
Step 4: Match with the given options:
This algebraic result is identical to option A, confirming the triangle-based answer without drawing any figure.
Final Answer:
The identity method gives the same result as the triangle method.
\[ \boxed{\sin(\tan^{-1}x)=\dfrac{x}{\sqrt{1+x^2}}} \]