Arctan of Product: How to Rewrite \arctan(a*b) Using the Product Rule?

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SUMMARY

The discussion centers on the mathematical expression \arctan(a*b) and the inquiry about a product rule for \arctan. Participants clarify that while there is no direct product rule for \arctan, the sum rule provided by Wikipedia can be utilized to manipulate expressions involving arctangents. The user also explores the possibility of simplifying the expression \arctan(\tan(A)B) through Taylor expansion, indicating a need for a method to express arctan when its argument is a sum.

PREREQUISITES
  • Understanding of trigonometric functions, specifically arctangent
  • Familiarity with Taylor series expansion
  • Knowledge of algebraic manipulation of expressions
  • Basic grasp of calculus concepts, particularly limits and continuity
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  • Learn about Taylor series and their applications in simplifying functions
  • Explore the sum and difference formulas for arctangent
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Students studying calculus, mathematicians interested in trigonometric identities, and anyone looking to deepen their understanding of arctangent properties and their applications in complex expressions.

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Homework Statement


Hi

Say I have an expression \arctan(a*b). Is there a product rule for \arctan that I can use to rewrite this? I tried the Wiki-page, but I couldn't find one there.
 
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There are some, Wikipedia lists
$$\arctan \alpha \pm \arctan \beta = \arctan\left( \frac{\alpha \pm \beta}{1 \mp \alpha \beta} \right)$$

Maybe you can ask a slightly more specific question (e.g. if you need this for a bigger problem, what is that problem)?
 
Thanks. OK, so after some calculations I get an expression of the form
<br /> \arctan(\tan(A)B)<br />
I wanted to Taylor expand B to first order in order to simplify, but I thought that would only make sense if there was a neat way to express \arctan when its argument is a sum.
 

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