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Math Help - Are there identities for inverse tangent?

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    Are there identities for inverse tangent?

    Are there identities for inverse tangent?

    Can you verify them [the identities] for me?

    I know two of them:

    arctan(x)=arccot(1/x)
    arctan(x)=pi/2-arccot(x)
    What is the third one if the identity exist? And Why?


    Thank you in advance.
    Last edited by Cbarker1; May 26th 2012 at 02:22 PM.
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    Re: Are there identities for inverse tangent?

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    Re: Are there identities for inverse tangent?

    Quote Originally Posted by Cbarker1 View Post
    Are there identities for inverse tangent?

    Can you verify them [the identities] for me?

    I know two of them:

    arctan(x)=arccot(1/x)
    arctan(x)=pi/2-arccot(x)
    What is the third one if the identity exist? And Why?


    Thank you in advance.
    For the first, let \displaystyle \begin{align*} y = \arctan{x} \end{align*}, then

    \displaystyle \begin{align*} \tan{y} &= x \\ \frac{1}{\cot{y}} &= x \\ \cot{y} &= \frac{1}{x} \\ y &= \textrm{arccot}\,{\frac{1}{x}} \\ \arctan{x} &= \textrm{arccot}\,{\frac{1}{x}} \end{align*}


    For the second, let \displaystyle \begin{align*} y &= \arctan{x} \end{align*}, then

    \displaystyle \begin{align*} \tan{y} &= x \\ \cot{\left(\frac{\pi}{2} - y\right)} &= x \\ \frac{\pi}{2} - y &= \textrm{arccot}\,{x} \\ y &= \frac{\pi}{2} - \textrm{arccot}\,{x} \\ \arctan{x} &= \frac{\pi}{2} - \textrm{arccot}\,{x} \end{align*}
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    Re: Are there identities for inverse tangent?

    why there is a pi-arctan(x)=arctan(x)+arctan(x/2)
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