Question

If \[x+\frac{1}{x}=2 \cos \alpha.\]Then Prove that;\[x^n+\frac{1}{x^n}=2 \cos n \alpha.\]

Answer 1

Let \[x=e^{i \alpha}\]&\[\frac{1}{x}=e^{-i \alpha}.\]Then \[[x^n+\frac{1}{x^n}]=[e^{i n \alpha}+ e^{-i n \alpha}]=?\]
Now wt to do?????

Answer 2

Looks like De Moivre to me.

Answer 3

So have I to expand it????????????????

Answer 4

\[ e^{i n \alpha } = (e^{i\alpha} )^n = \text{cis}n\alpha \]

Answer 5

I haven't tried yes @maheshmeghwal9 , but that's what I would try yes.

Answer 6

oh ok i gt it
thanx a lot:)

Answer 7

you are welcome