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Math Help - simplify

  1. #1
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    Smile simplify

    hello
    Can arctan(\frac{1}{\sqrt{1-x^{2}}}) be simplified ?
    thanks
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  2. #2
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    Quote Originally Posted by Raoh View Post
    hello
    Can arctan(\frac{1}{\sqrt{1-x^{2}}}) be simplified ?
    thanks
    Without further information, no it can't.
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  3. #3
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    Hello, Raoh!

    Can \arctan\!\left(\frac{1}{\sqrt{1-x^{2}}}\right) be simplified ?
    Well, we can make it "look nicer" . . . but that's all.


    We have: . \theta \:=\:\arctan\left(\frac{1}{\sqrt{1-x^2}}\right)

    . . Hence: . \tan\theta \:=\:\frac{1}{\sqrt{1-x^2}} \:=\:\frac{opp}{adj}


    So \theta is in a right triangle with: .  opp = 1,\;adj = \sqrt{1-x^2}

    Using Pythagorus, we have: . hyp \,=\,\sqrt{2-x^2}

    Then: . \csc\theta \:=\:\frac{hyp}{opp} \:=\:\frac{\sqrt{2-x^2}}{1} \quad\Rightarrow\quad \theta \:=\:\text{arccsc}\left(\sqrt{2-x^2}\right)

    Therefore: . \arctan\!\left(\frac{1}{\sqrt{1-x^2}}\right) \;=\;\text{arccsc}\left(\sqrt{2-x^2}\right)

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