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

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The discussion revolves around rewriting the expression \arctan(a*b) using a product rule, which is not readily available in standard references. Participants note that while there are formulas for the sum and difference of arctangents, a direct product rule does not exist. One user attempts to manipulate the expression into a form involving \arctan(\tan(A)B) and considers using a Taylor expansion for simplification. However, they express uncertainty about the feasibility of this approach without a clear method for handling sums within the arctangent function. The conversation highlights the need for more specific problem context to provide a better solution.
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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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Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

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