In an article "Generalized Reciprocals, Factors of Dickson Polynomials and Generalized Cyclotomic Polynomials over Finite Field" by Fitzgerald and Yucas I see on page 18 in the proof of Lemma 7.2 (2) mentioned a factorization formula by cyclotomic polynomials as $$x^n+1=\prod_{d|n,\frac nd \text{odd}}\Phi_{2d}(x)$$ This is new to me and probably I never saw it in the literature available to me.
Does anybody know where this formula is proved ?
I guess you are familiar with the formula $\displaystyle x^n - 1 = \prod_{d | n}\Phi_d$.
Noticing that $(x^n-1)(x^n+1) = x^{2n}-1$, you have $\displaystyle x^n+1 = \frac{x^{2n}-1}{x^n-1} = \frac{\prod_{d | 2n}\Phi_d}{\prod_{d | n}\Phi_d} = \prod_{d | 2n, d \nmid n}\Phi_d$.
Now $\{d, d | 2n \text{ and } d \nmid n \} = \{ 2d, d | n \text{ and } 2d \nmid n \} = \{2d, d | n \text{ and } \frac{n}{d} \text{ odd} \}$, which gives the wanted identity.