Given:
[tex]\sin(\tan^{-1}x)[/tex]To write:
The trigonometric function in terms of x.
Explanation:
Let us take,
[tex]y=\sin(\tan^{-1}x).............(1)[/tex]Let us assume that,
[tex]tan^{-1}x=p..............(2)[/tex]So, the function becomes
[tex]y=\sin p............(3)[/tex]From the equation (2),
[tex]\begin{gathered} x=\tan p \\ i.e)\tan p=\frac{x}{1}=\frac{Oppsite}{Adjacent} \end{gathered}[/tex]Using Pythagoras theorem,
[tex]\begin{gathered} Hyp^2=Opp^2+Adj^2 \\ Hyp^2=x^2+1^2 \\ Hyp=\sqrt{x^2+1} \end{gathered}[/tex]So, equation 3 becomes,
[tex]\begin{gathered} y=\sin p \\ =\frac{Opp}{Hyp} \\ y=\frac{x}{\sqrt{x^2+1}} \end{gathered}[/tex]Therefore, the composed trigonometric function in terms of x is,
[tex]\sin(\tan^{-1}x)=\frac{x}{\sqrt{x^2+1}}[/tex]Final answer:
The composed trigonometric function in terms of x is,
[tex]\sin(\tan^{-1}x)=\frac{x}{\sqrt{x^2+1}}[/tex]