Is there an identity for $\arctan(x+y)$?

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Ive tried looking on the internet and I can't seem to find any identities for $\arctan(x+y)$. I was wondering if anyone knows any?

Thanks!

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2 Answers

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The closest I know of is $$\arctan(x) + \arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$

(of course up to a multiple of $\pi$, and for $xy \neq 1$) Other than that I'm not aware of any useful ones.

Depending on what you want to achieve it might also be worth looking into Wikipedia:

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$$\arctan(x+y) + \arctan(x-y) = \arctan\left(\frac{2 x}{1-x^2+y^2}\right)$$

$$\arctan(x+y) - \arctan(x-y) = \arctan\left(\frac{2 y}{1+x^2-y^2}\right)$$

Taking sum

$$2 \arctan(x+y) = \arctan\left(\frac{2 x}{1-x^2+y^2}\right)+\arctan\left(\frac{2 y}{1+x^2-y^2}\right)$$

If $z= x+iy , u = $ real part of $ z^2 $

$$ 2 \arctan(x+y) = \arctan\left(\frac{2 x}{1-u}\right)+\arctan\left(\frac{2 y}{1+u}\right)$$

for whatever use it could be put to..

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