Find an argument for complex number $ \frac{1}{(1+2wi)^2} $

Question

Let $ w $ be a real number. Find an argument for

$$ \frac{1}{(1+2wi)^2} $$

Answer should be $ -2\arctan(2w) $. I keep getting $\arctan\left(\frac{-4w}{1+4w^2}\right) $ as an answer.

My steps involved multiplying both denominator and numerator with the conjugate of the denominator squared, which eventually led me to the answer above.

Answer 1

$\operatorname{arg}(z^m)=m\cdot\operatorname{arg}(z)+k\pi$ where integer $k$ such that $\operatorname{arg}(z^m)$ remains in $\in(-\pi,\pi]$

See this for the definition of arg$(x+iy)$

Here $\operatorname{arg}(1+2wi)^{-2}=(-2)\operatorname{arg}(1+2wi)+\pi$

$$\operatorname{arg}(1+2wi)=\arctan(2w)$$