<s>[INST] Proving identity $\frac{1}{x+iy}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}$ Below are some of the identities provided early in Needhams "Visual Complex analysis"
I want to verify the identity in my title.  From the identities given before it, I could only write 
$$\frac{1}{x+iy}=\frac{1}{x^2+y^2}\big(-\arctan\frac{y}{x}\big)$$
Not sure how to derive that $\big(-\arctan\frac{y}{x}\big)=x-iy$
thanks

 [/INST] $$\frac{1}{x+iy} \frac{x-iy}{x-iy} = \frac{x-iy}{(x+iy)(x-iy)} = \frac{x-iy}{x^2-i^2y^2} = \frac{x-iy}{x^2+y^2} = \frac{x}{x^2+y^2} - i\frac{y}{x^2+y^2}$$ as required
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