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

Let ${\sin}^{-} 1 x = \theta$ hence $x = \sin \theta$

For $0 < x < 1$ we draw a right triangle with hypotenuse equal to 1 and the other side equals to $x$ like the one in the Figure below.
From pythagorean theorem the other side is $\sqrt{1 - {x}^{2}}$

Now we know that

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\sin \theta}{\sqrt{1 - {\sin}^{2} \theta}}$

Because $x = \sin \theta$

We have that

$\tan \theta = \frac{x}{\sqrt{1 - {x}^{2}}}$

But from ${\sin}^{-} 1 x = \theta$ we get

tan(sin^-1x)=x/(sqrt(1-x^2)

Figure