Question

If $x=\cos \alpha +i\sin \alpha ,y=\cos \beta -i\sin \beta$ then $xy+\frac{1}{xy}=$

• A. $2\cos (\alpha +\beta )$
• B. $2(\cos \alpha +\cos \beta )$
• C. $2\cos (\alpha -\beta )$
• D. $2(\cos \alpha -\cos \beta )$

Answer 1

The correct option is C $2\cos (\alpha -\beta )$

We know $e^{i\theta }=\cos \theta +i\sin \theta$

$\therefore x=\cos \alpha +i\sin \alpha =e^{i\alpha }$

and $y=\cos (-\beta )+i\sin (-\beta )=e^{-i\beta }$

now we have, $xy=e^{i(\alpha -\beta )}$

similarly, $\frac{1}{xy}=e^{-i(\alpha -\beta )}$

So, $xy+\frac{1}{xy}=\cos (\alpha -\beta )+i\sin (\alpha -\beta )+\cos (-(\alpha -\beta \left)\right)+i\sin (-(\alpha -\beta \left)\right)$

$xy+\frac{1}{xy}=\cos (\alpha -\beta )+i\sin (\alpha -\beta )+\cos (\alpha -\beta )-i\sin (\alpha -\beta )=2\cos (\alpha -\beta )$