Real digital filter property
HINT:

From the definition of the DTFT

$$H(\omega)=\sum_{n=-\infty}^{\infty}h[n]e^{-jn\omega}\tag{1}$$

derive the following facts:


$H(\omega)=H(\omega+2\pi)$
$H(\omega)=H^*(-\omega)$ for real-valued $h[n]$


where $*$ means complex conjugation. Combine these two results to prove the given equation.