We define the even and odd parts of a complex sequence \(x(n)\) as

\[\mathcal{E}\{x(n)\}=\frac{x(n)+x^{*}(-n)}{2} \quad \text { and } \quad \mathcal{O}\{x(n)\}=\frac{x(n)-x^{*}(-n)}{2}\]

respectively. Show that

\[\mathcal{F}\{\mathcal{E}\{x(n)\}\}=\operatorname{Re}\left\{X\left(\mathrm{e}^{\mathrm{j} \omega}\right)\right\} \quad \text { and } \quad \mathcal{F}\{\mathcal{O}\{x(n)\}\}=\mathrm{j} \operatorname{Im}\left\{X\left(\mathrm{e}^{\mathrm{j} \omega}\right)\right\}\]

where \(X\left(\mathrm{e}^{\mathrm{j} \omega}\right)=\mathcal{F}\{x(n)\}\).