Question

If $z=\cos \theta +i\sin \theta$, then the value of $z^n+\frac{1}{z^n}$ will be:

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

Answer 1

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

Given, $z=\cos \theta +i\sin \theta$
$\therefore z^n+\frac{1}{z^n}={(\cos \theta +i\sin \theta )}^n+{(\cos \theta +i\sin \theta )}^{-n}$
$=\cos n\theta +i\sin n\theta +\cos \theta n-i\sin n\theta$
$=2\cos n\theta$