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

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May 26, 2018

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

enter image source here

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