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

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

We can use the principles of "SOH-CAH-TOA":

##tan^-1(x)=theta## is the angle when ##tan(theta)=x##.

Since ##tan(theta)="opposite"/"adjacent"##, we know that ##"opposite"=x## and ##"adjacent"=1##.

Using , we can see that the hypotenuse of a right triangle with legs ##x## and ##1## has ##"hypotenuse"=sqrt(x^2+1)##.

Now, to find ##sin(tan^-1(x))##, find ##sintheta## for the triangle where

##"opposite"=x## ##"adjacent"=1## ##"hypotenuse"=sqrt(x^2+1)##

Since ##sintheta="opposite"/"hypotenuse"##, we see that

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