Question

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

Answer 1

`\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}}`