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FP3 De Moivre's theorem

Can anyone help with this question?

ω = cos θ + i sin θ, show that (1 + &#969:wink:/(1 - &#969:wink: = i cot (θ/2)

(Which I've done)

Write down he roots of zn = -1, where n is a positive integer (also done). Hence writing z as (x - 1)/(x + 1) prove that the roots of (x - 1)n = -(x + 1)n are i cot [(2r + 1)Pi/2n] for r = 0, 1, 2, ..., n - 1.
Reply 1
Avatar for dvs
dvs
10
(x - 1)^n = -(x + 1)^n
=> ((x - 1)/(x + 1))^n = -1

Hence:
(x - 1)/(x + 1) = cos((2r+1)pi/n) + i sin((2r+1)pi/n)

Let w = (x - 1)/(x + 1). Then:
1 + w = (2x)/(x + 1)
1 - w = 2/(x + 1)
=> (1 + w)/(1 - w) = x

However, we also know that w = cos((2r+1)pi/n) + i sin((2r+1)pi/n), which implies that:
x = (1 + w)/(1 - w) = i cot((2r+1)pi/(2n))
Reply 2
Avatar for JCMaxwell
JCMaxwell
OP
Thank's so much; I don't think I would've noticed that. I would rep you, but the site won't let me :p:.

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