Question

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

Answer 1

Let `sin^-1x=theta` hence `x=sintheta`

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

`tantheta=(sintheta)/(costheta)=(sintheta)/(sqrt(1-sin^2theta))`

Because `x=sintheta`

We have that

`tantheta=x/(sqrt(1-x^2))`

But from `sin^-1x=theta` we get

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

Figure