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)##