# Algebraic rearrangement

Assuming that it is possible, how would you rearrange

$$x^{2n+1}-y^{2n+1}$$ as $$({\frac{x-y}{1+xy}})^{2n+1}$$?

I just need to do this as part of a larger proof. Some help would be appreciated! The $n$ relates to a series so the only variables that can change are $x$ and $y$. The proof is that

$$\text{arctan}(x)-\text{arctan}(y)=\text{arctan}\left(\frac{x-y}{1+xy}\right).$$

Instead of using identities I'm using the Maclaurin series for $\text{arctan}(x)$. So I need a way to transform the first expression into the second.

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It is often easier to find ways to prove two things equal than try to find a way to "transform" one into the other. – Hurkyl Apr 30 '14 at 8:06

For $n = 0$, the claim would be $$x-y = \frac{x-y}{1+xy},$$ which is false whenever $1+xy \not= 1$. If you want a counterexample for positive $n$, just take $x = 2$, and $y = 1$.