Question

If $x=\cos \theta +i\sin \theta$ the value of $x^n+\frac{1}{x^n}$ is

• A. $2\cos n\theta$
• B. $2i\sin n\theta$
• C. $2\sin n\theta$
• D. $2i\cos n\theta$

Answer 1

The correct option is A $2\cos n\theta$

$x=\cos \theta +i\sin \theta$

Applying Euler's form

$x=\cos \theta +i\sin \theta =e^{i\theta }$.

Hence

$x^n=e^{in\theta }$.

Similarly

$\frac{1}{x}=\overline{x}=\cos \theta -i\sin \theta =e^{-i\theta }$

Hence

$\frac{1}{x^n}=e^{-in\theta }$.

Hence

$x^n+\frac{1}{x^n}=e^{in\theta }+e^{-in\theta }$

$=\cos n\theta +i\sin n\theta +\cos n\theta -i\sin n\theta$

$=2\cos n\theta$.