Question

$(1+i{)}^n=2^{n/2}(cos\frac{n\pi }{4}+isin\frac{n\pi }{4})$

• A. True
• B. False

Answer 1

The correct option is A True

$(1+i{)}^n=(\sqrt{2}{)}^n(cos\frac{n\pi }{4}+isin\frac{n\pi }{4})$

We need to prove $(\frac{1+i}{\sqrt{2}}{)}^n=(cos\frac{n\pi }{4}+isin\frac{n\pi }{4})$

Taking RHS

$(cos\frac{n\pi }{4}+isin\frac{n\pi }{4})=e^{i\frac{n\pi }{4}}=(e^{\frac{i\pi }{4}}{)}^n=(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}{)}^n=(\frac{1+i}{\sqrt{2}}{)}^n$

Since, LHS=RHS

Hence, Proved.