Are there identities for inverse tangent?

**Cbarker1** · May 26th 2012, 02:18 PM

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.

**skeeter** · May 26th 2012, 02:55 PM

STPM Further Mathematics T: 5.1 – Inverse Trigonometric Functions

**Prove It** · May 26th 2012, 08:18 PM

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?

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

$\displaystyle \begin{align*} \tan{y} &= x \\ \cot{\left(\frac{\pi}{2} - y\right)} &= x \\ \frac{\pi}{2} - y &= \textrm{arccot}\,{x} \\ y &= \frac{\pi}{2} - \textrm{arccot}\,{x} \\ \arctan{x} &= \frac{\pi}{2} - \textrm{arccot}\,{x} \end{align*}$

**Cbarker1** · May 27th 2012, 08:46 AM

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

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