How do I simplify sin((tan^(-1))x)?

1 Answer

\frac{x}{\sqrt{1+x^2}}

Explanation:

Let

\tan^{-1}x=\theta\quad (\forall \ \ -\pi/2\le \theta \le \pi/2)

\tan\theta=x

\therefore \sin\theta=\frac{\tan\theta}{\sec \theta}

\sin\theta=\frac{\tan\theta}{\sqrt{1+\tan^2 \theta}}

\sin(\tan^{-1}x)=\frac{x}{\sqrt{1+x^2}}