Please, rewrite the expression as an algebraic expression in x. sin(tan^

tinfoQ 2021-08-22 Answered

Please, rewrite the expression as an algebraic expression in x.
\(\sin(\tan^−1x)\)

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Expert Answer

sweererlirumeX
Answered 2021-08-23 Author has 12633 answers

Let \(\displaystyleθ={{\tan}^{{-{{1}}}}{\left({x}\right)}}\), then \(\tan θ = x\)
Draw a right triangle with \(\displaystyleθ\) as one of it’s acute angles:
1) \(\displaystyle{A}{C}=\sqrt{{{A}{B}^{{2}}+{D}{C}^{{2}}}}\)
\(\displaystyle{A}{C}=\sqrt{{{x}^{{2}}+{1}^{{2}}}}=\sqrt{{{1}+{x}^{{2}}}}\)
2) \(\displaystyle{\sin{{\left({{\tan}^{{-{{1}}}}{\left({x}\right)}}\right)}}}=\frac{{x}}{{\sqrt{{{1}+{x}^{{2}}}}}}\)

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