Erroneous calculation using self-reciprocity of cyclotomic polynomials?

Question

Because of self-reciprocity of cyclotomic polynomials $\Phi_n(x)$ we have $$x^{\phi(n)}\Phi_n\left(\frac 1x\right)=\Phi_n(x)$$ with the Euler totient function $\phi(n)$. Now I concluded/calculated $$\prod_{d|n}x^{\phi(d)}\Phi_d\left(\frac 1x\right)=\prod_{d|n}x^{\phi(d)}\prod_{d|n}\Phi_d\left(\frac 1x\right)=x^n\left(\left(\frac 1x\right)^n-1\right)=1-x^n$$ and get on the other side $$\prod_{d|n}\Phi_d(x)=x^n-1$$ Sorry: I dont see what I do wrong.

Answer 1

The first equation doesn't hold for $n = 1$, for which $\phi(n) = 1$, $\Phi_1(x) = x - 1$, $\Phi_1(1/x) = 1/x - 1$, and $$ x^{\phi(1)} \Phi_1(1/x) = 1 - x \ne x - 1 = \Phi_1(x). $$

So there is a minus sign for $n \ge 1$, for $1|n$.