Given-that-e-i-npi-n-N-show-that-1-n-2-n-1-2-e-1-2-in-please-help-me-out-on-this-i-ve-stumbled-on-it-

Question Number 97413 by Rio Michael last updated on 08/Jun/20

Given that ω = e^{i θ}, θ \neq n π, n \in N

show that {(1 + ω)}^{n} = 2^{n} (\frac{1}{2} θ) e^{\frac{1}{2} (i n θ)}

please help me out on this, i^{'} ve stumbled on it .

Answered by mathmax by abdo last updated on 08/Jun/20

{(1 + w)}^{n} = {(1 + \cos θ + isin θ)}^{n} = {({2 \cos}^{2} (\frac{θ}{2}) + 2 isin (\frac{θ}{2}) \cos (\frac{θ}{2}))}^{n}

= {(2 \cos (\frac{θ}{2}))}^{n} {(\cos (\frac{θ}{2}) + isin (\frac{θ}{2}))}^{n}

= 2^{n} \cos^{n} (\frac{θ}{2}) {(e^{\frac{i θ}{2}})}^{n} = 2^{n} \cos^{n} (\frac{θ}{2}) e^{\frac{in θ}{2}}

there is a error in the Question!

Commented by Rio Michael last updated on 08/Jun/20

What a relief sir, i tried it out 10 times but {can}^{'} t

arrive at the required proof . Thank you so much .

Commented by mathmax by abdo last updated on 08/Jun/20

you are welcome .