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.
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 $\displaystyle \displaystyle \begin{align*} y = \arctan{x} \end{align*}$, then
$\displaystyle \displaystyle \begin{align*} \tan{y} &= x \\ \frac{1}{\cot{y}} &= x \\ \cot{y} &= \frac{1}{x} \\ y &= \textrm{arccot}\,{\frac{1}{x}} \\ \arctan{x} &= \textrm{arccot}\,{\frac{1}{x}} \end{align*}$
For the second, let $\displaystyle \displaystyle \begin{align*} y &= \arctan{x} \end{align*}$, then
$\displaystyle \displaystyle \begin{align*} \tan{y} &= x \\ \cot{\left(\frac{\pi}{2} - y\right)} &= x \\ \frac{\pi}{2} - y &= \textrm{arccot}\,{x} \\ y &= \frac{\pi}{2} - \textrm{arccot}\,{x} \\ \arctan{x} &= \frac{\pi}{2} - \textrm{arccot}\,{x} \end{align*}$