# Thread: Are there identities for inverse tangent?

1. ## 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?

3. ## Re: Are there identities for inverse tangent?

Originally Posted by Cbarker1
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?

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*}

4. ## Re: Are there identities for inverse tangent?

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