Question

If $x=cos\theta +isin\theta andy=cos\varphi +isin\varphi ,then{x}^m{y}^n+x^{-m}{y}^{-n}$ ,is equal to

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$x=cos\theta +isin\theta =e^{i\theta },y=cos\varphi +isin\varphi =e^{i\varphi }$
$\therefore x^m{y}^n+x^{-m}{y}^{-n}=e^{im\theta }{e}^{in\varphi }+e^{-im\theta }{e}^{-in\varphi }$
$=e^{i(m\theta +n\varphi )}+e^{-i(m\theta +n\varphi )}=cos(m\theta +n\varphi )+isin(m\theta +n\varphi )+cos(m\theta +n\varphi )-isin(m\theta +n\varphi )$
$+cos(m\theta +n\varphi )-isin(m\theta +n\varphi )=2cos(m\theta +n\varphi )$