text: # 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)
explanation: The document is a discussion thread about identities for the inverse tangent function. It starts with a user asking about the existence of identities for the inverse tangent and requesting verification of known identities. The user provides two identities and asks for a third one. Another user then provides a detailed verification of the first two identities. The thread ends with a question about a third identity involving \(\pi\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread about identities for the inverse tangent function. It starts with a user asking about the existence of identities for the inverse tangent and requesting verification of known identities. The user provides two identities and asks for a third one. Another user then provides a detailed verification of the first two identities. The thread ends with a question about a third identity involving \(\pi\).
  
  Label: multiple-question-and-answer