Good evening.
I'm currently working on cyclotomic polynomials $\Phi_n$, for $n \in \mathbb{N}^*$. I've proved by Möbius inversion formula that :
$$\forall x \in \mathbb{C}, \hspace{1mm} \Phi_n(x) = \prod_{d|n} (x^d-1)^{\mu(n/d)}$$
I would like to prove that if $n\in\mathbb{N}^*$ is odd, then :
$$\Phi_{2n}(x) = \Phi_n(-x)$$
The proof begins with :
$$\begin{array}{lcl}\Phi_{2n}(x) & = & \prod_{d|2n} (x^d-1)^{\mu(2n/d)} \\ & = & \prod_{d|n} (x^d-1)^{\mu(2n/d)} \prod_{d|n} (x^{2d}-1)^{\mu(2n/2d)}\end{array}$$
And I clearly don't understand why the second equality holds.
If someone could attempt to explain it to me I would be really grateful.