Finding the magnitude of a function

Question

Find $|X(\omega)|$ where $$ X(\omega)=\frac{1}{1-ae^{i\omega}} $$

Multiplication of the conjugate of the denominator on its numerator and denominator yields $$ X(\omega)=\frac{1-a\cos(\omega)-i\sin(\omega)}{1-2a\cos(\omega)+\omega^2} $$ which should give $$ |X(\omega)|=\sqrt{\frac{1-2a\cos(\omega)+a^2}{1-2a\cos(\omega)+\omega^2}} $$ However, the final answer on the book is $$ |X(\omega)|=\sqrt{\frac{1}{1-2a\cos(\omega)+\omega^2}} $$ Is the answer on the book wrong?

@Kavi Rama Murthy yes, my solution in the paper had that from the start. Thank you — wd violet, Oct 30 '21 at 12:44

score 1 · Accepted Answer · answered Oct 30 '21 at 12:39

$$X(\omega)=\frac {1-a\cos (\omega)+ia\sin (\omega)} {1-2a\cos (\omega)+a^{2}}$$ and $$|X(\omega)|=\frac {\sqrt {(1-a\cos (\omega))^{2}+a^{2}\sin^{2} (\omega)}} {1-2a\cos (\omega)+a^{2}}$$ $$=\frac {\sqrt {1+a^{2}-2a\cos (\omega)}}{1-2a\cos (\omega)+a^{2}}$$ $$=\frac 1 {\sqrt {1+a^{2}-2a\cos (\omega)}}$$

Finding the magnitude of a function

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