# Simplify \tan(\sin^{-1}(x))

Simplify $\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(x\right)\right)$
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Ben Owens
Let ${\mathrm{sin}}^{-1}\left(x\right)=\theta$, then $x=\mathrm{sin}\theta$
If we draw a right triangle with hypotenuse equal to 1 and the other side equals to x.
From pythagorean theorem: $\sqrt{1-{x}^{2}}$ is the other side.
Thus, $\mathrm{tan}\theta =\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }=\frac{\mathrm{sin}\theta }{\sqrt{1-{\mathrm{sin}}^{2}\theta }}$
As $x=\mathrm{sin}\theta$
Now, we have:
$\mathrm{tan}\theta =\frac{x}{\sqrt{1-{x}^{2}}}$
From ${\mathrm{sin}}^{-1}\left(x\right)=\theta$ we get
$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(x\right)\right)=\frac{x}{\sqrt{1-{x}^{2}}}$

Terry Ray
$\mathrm{tan}\left(\mathrm{arcsin}\left(x\right)\right)=\frac{x\sqrt{1-{x}^{2}}}{1-{x}^{2}}$