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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)##

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