Question

If ${\left[\frac{1+\cos \theta +i\sin \theta }{\sin \theta +i(1+\cos \theta )}\right]}^4$ $=\cos n\theta +i\sin n\theta ,thenn=$

• A. 2
• B. 3
• C. 4
• D. none

Answer 1

The correct option is C 4

$N^r=1+\cos \theta +i\sin \theta$
$=2\cos \frac{\theta }{2}[\cos \frac{\theta }{2}+i\sin \frac{\theta }{2}]=2\cos \frac{\theta }{2}.e^{i\theta /2}$

$D^r=-i^2\sin \theta +i(1+\cos \theta )=i\left(conjugateof{N}^r\right)$

$=i\times 2\cos \frac{\theta }{2}.e^{-i\theta /2}$

$\therefore E={\left(\frac{N^r}{D^r}\right)}^4={\left[\frac{1}{i}\frac{e^{i\theta /2}}{e^{-i\theta /2}}\right]}^4=\frac{1}{i^4}.e^{4i\theta }$

$=\cos 4\theta +i\sin 4\theta \therefore n=4$