# 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)
The text passage is a discussion thread about mathematical identities for the inverse tangent function. It includes a question about the existence of identities, verification of known identities, and a request for a third identity.
multiple_questions_with_answers

1. **question 1:** "are there identities for inverse tangent?"
   - **answer:** the text confirms the existence of identities for the inverse tangent function and provides verification for two of them.

2. **question 2:** "can you verify them [the identities] for me?"
   - **answer:** the text provides detailed step-by-step verifications for the two identities:
     - \( \arctan(x) = \arccot\left(\frac{1}{x}\right) \)
     - \( \arctan(x) = \frac{\pi}{2} - \arccot(x) \)

3. **question 3:** "what is the third one if the identity exists? and why?"
   - **answer:** the text does not provide a direct answer to this question. however, it mentions a potential third identity: \( \pi - \arctan(x) = \arctan(x) + \arctan\left(\frac{x
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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)
user
The text passage is a discussion thread about mathematical identities for the inverse tangent function. It includes a question about the existence of identities, verification of known identities, and a request for a third identity.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

1. **Question 1:** "Are there identities for inverse tangent?"
   - **Answer:** The text confirms the existence of identities for the inverse tangent function and provides verification for two of them.

2. **Question 2:** "Can you verify them [the identities] for me?"
   - **Answer:** The text provides detailed step-by-step verifications for the two identities:
     - \( \arctan(x) = \arccot\left(\frac{1}{x}\right) \)
     - \( \arctan(x) = \frac{\pi}{2} - \arccot(x) \)

3. **Question 3:** "What is the third one if the identity exists? And Why?"
   - **Answer:** The text does not provide a direct answer to this question. However, it mentions a potential third identity: \( \pi - \arctan(x) = \arctan(x) + \arctan\left(\frac{x