Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
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)##